There's a researcher named Alex Wellerstein who studies nuclear weapons history. He's done some AMAs here and posts on /r/askhistorians occasionally. He keeps a blog called Restricted Data. If you search around on it, there's lots of cool stuff. This post gets a little into your question. https://blog.nuclearsecrecy.com/2015/11/13/when-did-the-allies-know-there-wasnt-a-german-bomb/

Also, there's a BBC podcast called The Bomb. I don't think it was great, but there's enough useful info on it to make it worth listening to. But a few of the episodes address your question.