British and American POW's were treated as POW's. Soviet Jewish POW's were usually treated as Jews, if their national origin could be determined. The justification was that Soviet Union did not sign the international convention about POW's. Of course, this was the official point of view, but actual treatment depended on commanders in the field.
Official point of view was reflected in the so-called Commissar's Order issued before the invasion of Soviet Union. This order prescribed to select Commissars, Communists and Jews from POW's
and shoot them on the spot. But not all commanders obeyed this order. Some high standing German commanders refused to pass this order to the troops. As a result,
Jewish POWS were treated variously on Soviet theater.
Soviet citizens, except peasants carried the so-called "internal passport", where everyone was assigned a "nationality", which could Russian, Ukrainian, Jew etc. Military personnel did not carry this passport. So to determine that someone "is a Jew" was possible only from the words of other POW's, or checking for circumcision.
Corrections and references. The Commissar order is cited in this Wikiedia article:
https://en.wikipedia.org/wiki/Commissar_Order I do not know whether this citation is complete but it does not explicitly mentioned Jews.
In some documents they were disguised under the name "politically undesirable",
like in this original document:
which mentions "politically undesirable nationalities and racial groups".
What happened in reality is described in this Wikipedia article:
EDIT. There are many books which address the question, for example,
Alexander Dallin, German rule in Russia, 1941-1945: A study of occupation policies.