Wikipedia's List Of Cosmonauts page has a handy breakdown for "Soviet and Russian cosmonauts born outside Russia". According to that page:
All Soviet and RKA cosmonauts have been born within the borders of the U.S.S.R.; no cosmonaut who was born in independent Russia has yet flown. Many cosmonauts, however, were born in Soviet territories outside the boundaries of Russia, and may be claimed by various Soviet successor states as nationals of those states. All claimed Soviet or Russian citizenship at the time of their space flights.
As Heopps' answer notes, Ukraine and Kazakhstan have their own national space agencies, and have each flown one of their respective citizens.
It looks like 9 of the former SSRs bore a total of 35 cosmonauts:
- Azerbaidzhan S.S.R. / Azerbaijan: 1 (Musa Manarov, flew 1987 to Mir and stayed there for one year)
- Byelorussian S.S.R. / Belarus: 3 (First was Pyotr Klimuk, Soyuz 13, 1973)
- Georgian S.S.R. / Georgia: 1 (Fyodor Yurchikhin, flew on STS-112 to ISS in 2002)
- Kazakh S.S.R. / Kazakhstan: 6 (First was Vladimir Shatalov, Soyuz 4, 1969)
- Kirghiz S.S.R. / Kyrgyzstan: 1 (Shalizhan Sharipov, flew in 1998 (on the US space shuttle to Mir) and in 2004 to the ISS)
- Latvian S.S.R. / Latvia: 2 (First was Anatoly Solovyev, Mir, 1988)
- Turkmen S.S.R. / Turkmenistan: 1 (Oleg D. Kononenko, not to be confused with cosmonaut Oleg G. Kononenko (!), flew to ISS in 2008)
- Ukrainian S.S.R. / Ukraine: 19 (First was Pavel Popovich, Vostok 4, 1962
Georgy Beregovoy, Soyuz 3, 1968)
- Uzbek S.S.R. / Uzbekistan: 1? (Vladimir Dzhanibekov was born in Kazakh SSR, and flew to Salyut 6 in 1978 -- but his birthplace is now part of Uzbek SSR, so they might claim him.)
Note that after the dissolution of the USSR, citizenship in the successor republics was primarily based on residency, so most of the cosmonauts who were in the USSR's space program in 1991 wound up as Russian citizens regardless of ethnicity or birthplace.
Of some note is Aleksandr Volkov; born in the Ukrainian SSR (to an ethnic Russian family), he held USSR citizenship when he went to Mir for the second time in 1991, and during his stay, the USSR dissolved; he became a Russian citizen on his return. His son Sergey became the first second-generation cosmonaut in 2008.
Between 1978 and 1988, ten Soviet ally nations flew a total of eleven cosmonauts under the Interkosmos program, but these weren't member SSRs.
- Afghanistan: Abdul Ahad Mohmand — Soyuz TM-6/5, 1988
Aleksandar Panayotov Aleksandrov — Soyuz TM-5/4, 1988;
Georgi Ivanov — Soyuz 33, 1979
- Cuba: Arnaldo Tamayo Méndez — Soyuz 38, 1980
- Czechoslovakia: Vladimír Remek — Soyuz 28, 1978
- East Germany: Sigmund Jähn — Soyuz 31/29, 1978
- Hungary: Bertalan Farkas — Soyuz 36/35, 1980
- Mongolia: Jügderdemidiin Gürragchaa — Soyuz 39, 1981
- Poland: Mirosław Hermaszewski — Soyuz 30, 1978
- Romania: Dumitru Prunariu — Soyuz 40, 1981
- Vietnam: Phạm Tuân — Soyuz 37/36, 1980