Oppenheimer was an American, so that is just a misleading association (his father was a German, and emigrated to the US in the 19th century). But the others were certainly refugees from Europe, and there were several dozen of them on the Manhattan Project, many from Axis nations (we could include Enrico Fermi in that, as well).

As for why — because they were some of the top scientists of their time and were fervently anti-Nazi. They were in the US to escape the Nazis and were among the most motivated to stop them, because they had seen first hand what the threat was. European refugees, especially those of Jewish descent, were among the most important instigators in the nuclear programs of the United States and United Kingdom for exactly this reason: they had more to fear than the others, at least early on.

Were they totally trusted? No, not at all. Many of them were classes as "enemy aliens" in the United States and there were tight restrictions on their ability to travel. Some of them had difficulty getting sufficient security clearances to participate in war work because the Army and FBI were suspicious of them. Those that were allowed to work on the project were people whose anti-Fascist credentials were considered sufficiently compelling, and whose colleagues were willing to testify about their reliability. But there was, especially early on, a lot of suspicion from the military. All of those who were not associated with the British project (like Fuchs) became US citizens as part of the process of showing their new loyalty.

Those in charge of the actual bomb project, however, were so interested in the expediency of the thing — actually making the bomb on time — that they were willing to cut some corners on background checks and things like that. The main problem here ended up not being Axis agents, but Communists. Fuchs, for example, turned out to be a Soviet spy. An anti-fascist, for sure! But a security risk of a different sort. Oppenheimer himself had sufficiently dubious political associations from the 1930s to fall under suspicion, but the concerns of the security forces were overruled by their superiors on the project specifically because he was so important to accomplishing the ultimate goal.

(To say all of the above is not to imply, in any way, that the approach taken with Japanese-Americans was at all justified relative to the treatment of these scientists. I am sure there were Japanese-Americans who could have meaningfully contributed to the Manhattan Project in one form or another had they been given the chance. They were handled quite differently by the federal government, as is obvious.)