What did Alan Turing have in mind when he conceived his universal computing machine?

One could speculate that his train of thoughts was like this:

- I can simulate any of my machines (there’s experimental evidence, and of course I defined them to work like people doing maths on a piece of paper).
- I’ve already formulated a computability thesis that says: if I can do it, then a computing machine also can.
- But then, there must be a universal computing machine! Let’s fill in the details…

Or maybe he came to that realisation some completely different way. Possibly, directly from Gödel’s work?

Who knows, maybe there even exists something written by Turing himself on this subject.