The war made any effort to rescue Jews more complicated. The war quickly became a struggle for national survival in many countries. This became true most of all for Britain and then for the Soviet Union. American entry into the war at the end of 1941 held out the prospect of eventual Allied victory. It did not bring about actual victory immediately. National interest came before any other interest in the allocation of resources and the planning of operations.
Sometime during the Winter of 1940-1941, Adolf Hitler made two decisions. First, he would attack the Soviet Union in Summer 1941. Second, he would have all the Jews of Continental Europe killed as quickly as possible beginning in Summer 1941. The armed forces got busy preparing the attack on the Soviet Union. The SS got busy preparing the annihilation of the Jews. Both decisions were highly-guarded state secrets.
The British and the American leaders learned of the attempt to annihilate—rather than persecute—the Jews in the second half of 1942. They had some difficulty comprehending what they were told. Felix Frankfurter said later of one messenger from the abyss that “I did not say that he was lying, I said that I could not believe him. There is a difference.”
Then there was the problem posed by the speed of the killing. “In mid-March 1942, some 75 to 80 percent of all victims of the Holocaust were still alive, while 20 to 25 percent had perished. A mere eleven months later, in mid-February 1943, the percentages were exactly the reverse. At the core of the Holocaust was a short intensive wave of mass murder.” Thus, 50 to 60 percent of the killing took place in less than a year. Before Stalingrad, before El Alamein, before “Torch,” and before Americans began bombing Germany.
At a rough estimate, by the end of 1941, the Germans had under their direct control 5.7 million Jews. There was nothing that could be done for these people by outsiders. The Germans were determined to catch and kill as many as possible.
In late 1942, how many Jews were outside the direct control of the Nazis? Something on the order of 1.3 million. There were 48,000 in Italy, 48,500 in Bulgaria, 445,000 in Hungary, and 756,000 in Rumania.
The Italian government resisted deportations until the Germans invaded Italy. When the Germans occupied northern Italy in 1943, there were 43,000 Jews in their area. An estimated 35,000 survived (81%). Most Italians refused to cooperate with the Germans.
There were 48,500 Jews inside Bulgaria’s borders of 1939. An additional 11,000 were added in territory seized from Greece in 1941. The Bulgarians turned over the Greek Jews to the Germans. Virtually none survived. The Bulgarian refused to turn over Bulgarian Jews.
There were 756,000 Jews inside the Rumanian borders of 1930. An estimated 490,000 (65%) survived. There were an additional 185,000 Jews in Bessarabia and Bukovina, taken from the Soviet Union. An estimated 55,000 (30%) of these survived. The Rumanian government obstructed deportations from its core territory.
There were 825,000 Jews inside the Hungarian borders of 1941. An estimated 261,000 survived (31%). The Germans had to overthrow the Hungarian government to get at these Jews.
 Walter Laqueur and Richard Breitman, Breaking the Silence (1986).
 Christopher Browning, Ordinary Men: Reserve Police Battalion 101 and the Final Solution in Poland (1992), p. xv.
 The numbers get a little murky because of wartime changes in boundaries. Hungary, Rumania, and Bulgaria all added territory with Jewish populations.
 For all figures see: Jewish Losses during the Holocaust: By Country | Holocaust Encyclopedia (ushmm.org)