As per my personal experience, mail was vigorously censored in the period after WW2 until mid/late sixties??? During Glasnost there were no problems communicating by mail.
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Intl Privacy Laws are working against your success.

It would be more effective if the family in Ukraine made the inquiries for themselves.

Contacting the Tracing Service & Family Reunification Division of the Intl Red Cross in Washington DC is a gd bet. Searches are lengthy and can take up to three yrs.

There are Red Army Conscription records online.
These records are in Russian & will confirm family
history re relative's disappearance.
http://www.obd-memorial.ru/

Since their relative landed up in USA, he might have been listed in the records for DP (Displaced People) in post WW2 Germany. Turn around time can take as much as three-six months.
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