It depends on your definition of "computer".

There was a period of very rapid development, largely government funded efforts as both sides of the war saw computers as critical strategically, and a bunch projects went from "hey do you think this might work?" to "here's an unlimited budget, go make it work."

They were all heavily influenced by each other (and spying on each other, and lying about the extent of their intelligence gathering capabilities) and computers were progressively developed in paralel.

Who did it "first" depends on where you draw the line in the sand and say "yes, this is a computer". Even the "turing" test doesn't work as a clear definition, because the first computers that could pass the that test were barely able to pass in practice.

Also, I think you could make a compelling argument that none of those projects would've received all that funding (and there definitely would've been less espionage) unless a war was going on. If the war hadn't happened, computers would've taken much longer to be invented.