I have some numbers on hand from Velikaya Otechestvennaya Voyna bez grifa sekretnosti (roughly Great Patriotic War declassified) by Krivosheev and Adronikov. Unfortunately they do not report on the demographics of the entire Red Army, but they do give enough data to demonstrate that far less than 90% of Soviet soldiers were Russian.

Out of 8,668,400 total combat losses where nationality could be established, only 5,756,000 were Russian (66%). Ukrainians made up 1,377,400 losses (16%), Belarusians 252,900 (3%), Tatars 187,700 (2%), Jews 142,500 (1.6%), Kazakhs 125,500 (1.4%), Uzbeks 117,900 (1.4%) and other nationalities made up less than one percent of the total losses apiece.

The number of prisoners of war tallied also show that Russians were the most numerous population involved, but that much. Out of 1,368,849 cases tallied up in the book, 657,339 POWs were Russian (48%), 386,568 were Ukrainian (28%), 103,053 were Belarusian (7.5%), 30,698 were Tatars (2%) 28,228 were Uzbeks (2%), 23,816 were Georgians (1.7%), 23,142 were Kazakh (1.7%), 20,850 were Azeri (1.5%), 20,067 were Armenians (1.5%) and then the remaining nationalities made up less than 1% each.

When determining the percentages of each nationality that fought in the Red Army it is also important to remember that pre-war populations won't do you much good. Ukraine, Belarus, and the Baltics were fully occupied by the Germans for the majority of the war, meaning that any civilians who would come of fighting age during this time were unable to join the Red Army. Men of fighting age or nearing it would likely be kidnapped for forced labour in Germany or killed, which further drove down those numbers. In contrast, it would be much easier to draft someone who turned 18 in Eastern RSFSR, Kazakhstan, Uzbekistan, etc. so the proportion of these nationalities were higher.