Hard to give an actual date, but the process came to a head some time in mid 1964.
I took a second look inside Mike Lang´s book for this, but did not find explicit mention. Unfortunately his narratives covering the fifties and sixties are much less comprehensive (and probably also more reluctant to name names) than for the eighties. So the only passages I found by reading between the lines are the follwoing occasions:
Dutch GP 1964 (the organizers were always notorious for being quite selective in their selection of entrants):
"BRP were absent because of an acute shortage of cars, but apart from them all the other leading teams were present and supported by the more regular private entrants who had been invited to make up a field of eighteen." Ok, the BRP cars had damage at Monaco, but a fourtnight should be enough for example to fix a leaking tank on Taylor´s car for example. So did they really have a shortage of cars? Sheldon says, it was decision by the organisers not to invite BRP, which would have been also very strange regarding the presence of Anderson and Siffert with their private Brabhams.
The BRP were missing also at the German GP, but without explicit mention by Lang. But Sheldon writes "The British Racing Partnership were unable to come to terms with the organisers about their worth so forfeited their entry." Seems there could have been quite fundamental differences between the AvD and Ken Gregory, as for 1962 I have found almost similar words in Sheldon´s text: "Only the British Racing Partnership were absent, the team had a difference of opinion with the organisers as to the amount fo starting money they were worth"
Other mentions of starting money / number issues during the early sixties:
1960 Monaco GP
Lang: "Although 24 entries had been accepted the number of starters was being limited to 16 again."
1960 Dutch GP
Lang: "...but in the background the entrans were arguing with the officials over the qualifying conditions. Eventually it was agreed that the original number of starters permitted be increased from fifteen to twenty but no starting money would be paid to the extra five. With this settled the air became a bit clearer though there was still some discontent amongst the teams who were unlikely to be in the money."
Sheldon: "The organisers of the Dutch GP did their best to cause disharmony among the ranks by inviting 21 cars but saying that, although everyone could start, only the best 15 in practise would be paid starting money. The upshot was that, although Chuck Daigh qualified, Reventlow withdrew both cars, Salvadori´s Aston Martin was withdrawn, Gregory withdrew but Trintignant came to a special deal with the organisers. Godin de Beaufort, being Dutch was going to start any way. [...] The start was rather funny because Trintignant´s deal involved his getting into the first ten places in the early stages of the race. Only then would he qualify for starting money."
1960 Portuguese Grand Prix
Sheldon: "The organisers only invited the cream of the Grand Prix world but gained a drive for their man, Cabral, in the Centro Sud Cooper."
1961 Monaco GP:
Lang: "As usual though the number of permitted starters was being restricted but this year each of the five 'works' teams were guaranteed a place on the grid for theior two leading drivers as were Moss and Trintignant for being past winners. This left only four places for the remaining nine drivers present to fight over."
Sheldon: "The organisers only accepted a certain number of potential qualifiers which meant that, for instance, the second Reg Parnell Racing entry was disappointed."
1961 Dutch GP:
Lang: "This year the entry list was made up of two cars from each of the 'works' teams except Ferrari who had three, one car from Rob Walker and Yeoman Credit, two Porsches entered by the Durtchman de Beaufort and finally a Lotus and Cooper from Camoradi International, the last two cars being accepted as reserves, which, in the event, became non-starters."
Sheldon: "The organisers invited exactly who they pleased. All the factories were allowed two entries. Ferrari were allowed an extra one to balance the British teams while Porsche managed four by virtue of the local driver borrowing two cars from the works and loaning one of them to Herrmann. Gregory and Burgess were invited as reserves but with no hope of a start unless one of the main invites failed to make the grade."
1961 Belgian GP:
Lang: "Although the organisers had invited twenty-five entries, only a pre-selected sixteen plus the fastest three out of the other nine were to receive starting money. It was then a matter of personal choice for the six unlucky ones to decide whether or not to start in the race. Included in the nine who had to qualify was the UDT-Laystall team [BRP!] with their Lotus and, having two drivers present, they decided to let Allison and Taylor both practise, the faster earning the right to drive in the race."
1961 French GP
Lang: "With the organisers making no restrictions about who could start and who couldn´t the large field of 26 cars lined up on the grid..."
German GP
Sheldon: "...BRP giving the race a miss because of a commitment to an American race."
1961 Italian GP:
Lang: "A huge entry of 37 had been accepted and no restriction was being made to the number of starters providing each individual driver recorded a time no more than 15% slower than the second fastest practise time."
1962 Dutch GP
Lang: "..although the entry was restricted to twenty, all the leading teams were present."
Sheldon: "...the Dutch organisers, as was their wont, selcting the field by personal whim rather than by logic. This meant that such worthies as Lewis and Marsh were ignored while Ben Pon and the sluggish Seidel both got drives. In the end Lewis was invited because no car was available for Trintignant..."
1962 Monaco GP
Lang: "Like the previous year, the organisers invited two entries from each of the five 'works' teams leaving six places available for the remaining eleven entries present to fight over. These included Brabham and Trintignant, no concession being made to past winners this year."
Sheldon: "if the metho of invitation that the organisers of the Dutch GP adopted was silly then that of the Monaco organisers was equally dotty. With only 16 starters allowed, two entries from each of the five manufacturers were given a firm start while everyone else had to qualify for the six remaining places."
1962 British GP
Sheldon: "#52 Jo Siffert dna, not enough starting money offered."
1962 Italian GP
Lang: "... the organisers had decided, however, to allow only 22 cars to start in the race subject to the proviso that each individual driver recorded a time within 10% of the second fastest practise time."
Sheldon: "The only absentee from the entry list was Brabham who could not come to terms with the organisers in respect of starting money."
1962 South African GP
Sheldon: "The entry was similar to the two previous races with the more uncompetitive locals not being accepted."
1963 Monaco GP
Lang: "Altogether twenty-four entries were invited to practise for this, the first Championship event of 1963, even though only sixteen cars would be allowed to start in the race as had become usual. Perchance, though, only sixteen of the entries materialised so the decision by the organisers to guarantee starting places only to past winners or to the current and former World Champions proved incidental."
1963 French GP
Sheldon: "Arundell was prevented from starting for he was racing in the Formula Junior race and it was now not permitted to drive in another race within 24 hours of competing in a Grand Prix"
1963 German GP:
Lang: "When practise times were sorted out the organisers decided that Pilette, Raby, Parnell and Kuhnke had been too slow and restricted the number of starters to twenty-two."
Sheldon: "The organisers of the German Grand Prix, like those of the British race invited everyone who wanted to come but stipulated that only a certain number of the also-rans would get a start thereby eliminating some of the really slow drivers like Kuhnke and Parnell."
1963 Italian GP
Lang: "Throughout the afternoon most of the others were having to work hard in order to try to qualify for the organisers were accepting no-one who failed to achieve a time less than 10% slower than the second fastest practise time. Even then only 20 starters were being allowed to start in the race which meant that eight of thos epresent would be unlucky."
1963 South African GP
Sheldon: "The organisers invited all teh European factory teams plus one private entry. As well as this dear Godin de Beaufort turned up in the hope of a drive. Luckily for him Mike Hailwood did not arrive so he was given a drive prviding he lapped in 1:37,9 - over two seconds better than he managed last year."
1964 Monaco GP
Lang: "When all times had been collated Amon, Revson and Collomb were the unlucky ones to be excluded from the race, only the fastest 16 being accepted."
1964 Dutch GP
Sheldon: "As usual, the organisers of teh Dutch Grand Prix invited just whom they chose, to drive in their race. In essence this was the usuals plus de Beaufort and minus the British Racing Partnership."
1964 French GP
Lang: "Apart from Scuderia Centro-Sud and Rob Walker the organisers attracted all the regular entrants..."
Sheldon: "Considering that there was no particular restrictions put on entries there was the smallest turnout of cars for some time."
1964 German GP
Sheldon: "#24 and #25 BRP entries: dna, disagreement with organisers."
1964 Italian GP:
Lang: "Once again however the total number of starters was being restricted to twenty so with twenty-five entries materialising everyone was anxious to make an early start when the firts practise session opened..."
1965 South African GP
Lang: "This did nothing to prevent a large entry from materialisng, however, but after allowing for guaranteed places on the grid for drivers of the regular teams, one for Frank Gardner in the John Willment team and another for South African champion John Love, only four places were left for the other drivers present. To sort things out a bit a special pre-practise eliminiation session was held. Out of the twelve though, only three failed to lap under the prescribed time of 1 min 37.0 sec..."
Sheldon: "The local drivers had a special pre-qualifying session. They had to lap in under 1:37 in order to qualify for practise and three drivers failed to do this. Once these drivers had pre-qualified they had another hurdle to mount for only 20 cars would be allowed in the race."
1965 Monaco GP
Lang: "To support Spence at Monaco, Team Lotus had entered Rodriguez but after a wrangle with the organisers over starting money both entries were withdrawn"
Sheldon: "The organisers, when they heard of Clark´s absence, only guaranteed Lotus one starter so they withdrew their entries. There were therefore 17 drivers competing for the 16 starting positions..."
1965 Belgian GP
Lang: "Proceedings got off to a bad start for this year´s European Grand Prix due to wrangles over starting moeny. The crux of the matter was that this was being paid only to sixteen starters. Of course two drivers from each of the 'works' teams were guaranteed places on the grid and consequently starting money leaving just four places for ten privateers who had been invited. Fortunately this resolved itself by the organisers agreeing to pay starting moeny to a further two cars and as the entries of Hawkins and Mairesse failed to materialise only the two slowest would be unlucky. Nevertheless agreement was not reached until after the first practise session so only the twelve 'works' drivers took advantage of the 2 hours provided on Friday evening."
Sheldon: "There was almost a strike before the Belgian GP. The aggrieved people were the private entrants. The organisers invited ten drivers but when they arrived, they found that only four would be paid starting money. They therefore abstained from the first session. The organisers relented and, out of the eight who arrived, six would be paid with thze other two being allowed to start but not be paid. In the end only Anderson was left out, having problems he could not fix in order to set a good time. Gregory opted to start without starting money."
1965 French GP
Sheldon: "The organisers tried to persuade the Willment team to provide a car for Jo Schlesser. When all that was produced was a Formula 2 Brabham, they cancelled not only that entry but Gardner´s as well. Anderson was cabled and rushed down to take part."
1965 British GP
Lang: "Ith the organisers calling for a lap within 5 seconds of third fastest time in order to qualify there was no lack of interest among the slower drivers."
Sheldon: "In addition to the usual runners there were a few who were not normally seen at Grand sPrix. They knew that the organisers were prepared to accept anyone provided they could lap a time not greater than 110% of the pole-sitter."
1965 Dutch GP
Sheldon: "The organising club had no truck with qualifying and only invited 16 cars to take part. Honda withdrew an entry and the club increased the field to 17 by extending invitations to a second Parnell car and Bob Anderson."
1965 German GP
Sheldon: "The long length of the Nürburgring put no constraints on the number of starters in the German GP providing that drivers could lap within 110% of the pole position time - a new requirement for Grands Prix."