it does seem like working on the Bombe got Alan Turing really deep into thinking about hardware. he'd previously played around with building some logic circuits but seems like he didn't go deep into it before Bletchley, before that he was mostly a very pure mathematician.
Turing was interested in lots of things. As a coincidence one of the small jobs I did many years ago was to develop the Turing Archive's web site, which I'm glad to see has since been substantially renovated: https://turingarchive.kings.cam.ac.uk/ - For that work we had a lot of high resolution images of non-Computing stuff Turing cared about, including Morphogensis (basically, why things are the shapes they are, for example why is a rose petal shaped that way? Why stripes on giraffes but spots on dalmations?).
So mathematics yes, Turing was always interested in that, but always applicability was on Turing's mind. In the early twentieth century Turing's "machine" in a paper he wrote at Princeton (before the war) was just an idea, but er, obviously with the exception of the need for an "infinite" paper tape you can realise Turing's machine, it's a computer, familiar to everybody today.
Even after Bletchley Turing wasn't a Software Engineer. The meta-applicability isn't something which would occur to a mathematician. That took Grace Hopper. Grace understood that the problem her people were tasked with (mechanically convert instructions to the code for the actual machine) is exactly the same sort of task the machine is doing anyway, and she invents the Compiler.
