As far as concerned post-war races...
... After the WWII in Soviet Union was taken (as war trophies ;-) 18(eighteen) A-U, a few Alfa-Romeos 6C-250CC, 8C-2900C, Jaguar CC100, BMW-328, Wanderer W25K and other sport cars. But, despite close attention of constructors and engineeres to those cars the results of their work remains only on papers and had no influence on real soviet cars. One of the sad reasons for that was the unsuccesful record breaking attempt, organized in the summer of 1948 near town Gorky (today - Nizhnij Novgorod). The streamlined Auto-Union crashed havily during the trial and killed 18 people. In december 1948 with the special decree Soviet goverment banned the foreighn technick from all auto & moto competions. The decision was proved by the stupid and demagogical idea about the nessesity of the development our own (read - soviet) racing cars, and not used the allien 'capitalist' experience.
The first USSR championship was held near Moscow in the beginning of August 1950 on the Moskow-Minsk highway. There was only one 300-kms race (so called "linen race": 150-km in one way and forth) in which 45 cars was entered (43 participated). The next year also 84-kms auto-cross was included in national championship programme.
(BTW, before the first official Soviet ch. there are a lot of local championships (in Moscow, Leningrad, Minsk (Belorussia), Estonia, Latvia, etc.)
1950 USSR ch.
(up to 2500cc)
1. V.Metelev-V.Rodionov (Gorkij) Pobeda-Sport
2. N.Sorokin-A.Shalashov (Gorkij) Pobeda
3. K.Nikishin - E.Beliakov (Gorkij) Pobeda
(up to 1200cc)
1. Givartovskij-A.Kokorev (Moscow) Moskvich-403 E(Exsperimental)424
2. M.Tourkov-A.Glazov (Moscow) Moskvich-400
3. N.Kachigin-N.Shkaldanov (Moscow) Moskvich-403-E424
First USSR Champ Mikhail Metelev
another 1950 champions - Lev Givartovskij and his mechanic Kokorev
The same as Metelev's winning car - Pobeda-Sport. Only three cars were built.
more powerful 1951 version. Max speed approx. 162 km/h.
About russian Automobile federation: from october of 1956 CAMK (Central Auto-Motor Club of USSR), ruled by military organisation DOSAAF (today ROSTO) was acclaimed as the full member of F.I.A. Till today the CAMK are one of the key persons in Russian motorsport. But in Soviet times the Automobile Federation of USSR (FAS) was only the executive representative of the weel of CAMK-DOSAAF (read - Soviet Army).
... about Soviet F1-cars (simply don't forget that this is so-called F1 cars). So... there are NO soviet F1 driver till today.
P.S. on Rainer's photo (naturally from the L.Shugurov book about soviet autosport) there is the 1958 USSR champion (class "G" - "gonochnije", i.e. racing cars of all types with handicap) Valery Shakhverdov at the wheel of self-built GA-22 (engine GAZ-21 "Volga"):
http://w1.859.telia.com/~u85900964/AtlasBB/GA22-1960-02.jpg
P.P.S. For those who are more curious you can see a bit more results (champions of the USSR) here:
http://www.geocities.com/flicker_viv/chUSSR-1.html
(sorry... now only in Russian)