The Colossus Mark 1 was the first computer to be both programmable and have conditional branches. The previous machines you mention were not computers by any modern definition.
C.
If you want to play that game, ENIAC was the first to be turing complete and that is US based. Turing complete is important to modern computing.
Z3 was in fact programmable and had conditional branches, it was just electromechanical.
I think it's safe to say pretty much everyone at the time was working on it and no one was really first to be there. It's something that will always remain a biased debate.