There is a section on this question in the FAQ. The very short answer is, they weren't close at all, because they weren't actually running a bomb development project. They had a very modest reactor research project, and it did not achieve a working reactor by the end of the war. They were, in other words, years away from producing nuclear weapons.

As for whether they could use a weapon they might have magically come upon, it would depend on what "use" meant and the dimensions of whatever weapon they might have had. They did not have the means to bomb the United States, conventionally or otherwise. Could they have detonated one on their own soil if they had one? Sure. Could they drop on on London? It would depend on the size and weight of the weapon, but if they were akin to the US weapons developed, possibly.