Correct. Turing worked in the US at the end of the war and they got all his knowledge. When we destroyed all the bombes, they didn't.I think they do pet IIRC. I think there was a brief mention.
I think, but @HellsBells may know better than me as she's doing a lot of reading about it, that it was actually Colussus and the cracking of Tunny (a different encryption system from Enigma) rather than Turing's Bombe which cracked Enigma that led directly to the development of the computer.
Also I think I'm right in saying that we shared the technology with the Yanks as part of the deal for all the £££ they gave us in the war, and partly because that's what allies do of course. The Yanks then took the technology and saw how it might have peaceful/commercial application, and that led directly to IBM, and to Microsoft, and to Apple. Meanwhile, our government insisted that even after the end of the war it remained ultra ultra secret, so no research into the commercial possibilities was ever done. This is why Silicon Valley is in California rather than in Buckinghamshire: we invented the fucker but failed to exploit it. I think I'm right in saying this but may be wrong.
Edit: on the Tunny bit, Tommy Flowers invented that machine to crack a different machine that the Germans were using - the Lorenz machine. TheTunny machine was the worlds first digital programmable computer.
I'm loving reading all of this shit like, it's great.