The Trinity Test, where a prototype bomb was detonated to essentially serve as a proof of concept, only occurred on July 16th 1945.

It was only then that the ongoing work of producing usable bombs was proven usable really(The Gadget detonated here was the same design used in Fat Man the Nagasaki bomb), while everyone was pretty sure it would work ahead of time, you still never quite know and nobody had ever set off a nuke before so a test was deemed desirable before active use.

Germany had of course surrendered 2 months before then.

The European campaign was already over before the Nuclear bomb was finished. The European war was over on the eighth of May 1945, the war in the Pacific was over on 15 of August 1945. The Nuclear bomb was not finished until sometime between the two dates.