Alas, the computers they were able to build at that time were neither fast enough nor big enough for it to work as you're imagining. Bletchley Park had hundreds of employees trying to decode messages through the end of the war. They wouldn't have had a prayer without Turing, but what he built was a set of mathematical and hardware tools that allowed a huge staff of codebreakers to try to figure out each day's code. (The Germans changed the settings on the Enigma every day.) Once they had the day's code, the rest of that day was easy. But there were plenty of days when they didn't break the code at all.
(I've just finished reading The Enigma Girls, a book for teenagers about ten of the (upper-) teenage girls who worked at Bletchley, who published memoirs very late in life because the UK Official Secrets Act forbade them from ever telling anyone what they had done, until finally the government declassified the Bletchley operation.)
PS They also wouldn't have had a prayer without the early work of Polish mathematicians, whose names nobody remembers including me, although they did the hardest part of the job, analyzing how the Enigma worked in the first place and making mathematical models of it. (Yeah, I'm sure somebody remembers, probably including Wikipedia, but I mean no non-experts know anyone's name but Turing.)
Alas, the role of the US in this story is quite limited. Quite late in the war, they built a faster computer for other purposes (hint: engineering the atomic bomb took a lot of applied math) and let the Bletchley team use it also.
What makes Turing and Church stand out from Babbage and all those other computing pioneers is that they founded computer science, the theoretical understanding of computation. They published independent solutions to Hilbert's Entscheidungsproblem (decision problem), which asked for an algorithm to determine whether any given statement is provable in logic. Church proved that it's impossible to determine, in general, whether any two procedures compute the same function; Turing proved that it's impossible to determine in guaranteed finite time whether a given Turing machine will halt in finite time for a particular input tape.
We Lispians consider it terribly unfair that almost everyone credits Turing alone with the solution to the Entscheidungsproblem even though Church actually got there slightly earlier. I think this happened because everyone recognizes a Turing machine as an algorithm, whereas back then nobody understood what Church was talking about, or how functions embody algorithms. It was only when McCarthy turned Church's ideas into an actual computer programming language (Lisp) that people got it -- and not all people, for half a century after that. (I should emphasize that Turing himself did understand and acknowledge that Church got there first.)
(By the way, one of the things Snap! users ask for periodically is that = should report True for two procedures that do the same thing. What Church proved is that that's not possible in general.)