I can't speak for Turing and I imagine every genius' mind works somewhat differently.
In the early days of Bletchley Park, Turing was charged with organizing the efforts of many human "computors" (i.e., people who performed some computation) who sat in ranks at long tables in one of the buildings. The idea was to break the task (decryption) down into tiny operations in order to speed things up and also to reduce errors. IIRC, Turing organized the computations on color-coded cards where the color was a primeval "program" that guided the card through the maze of human computors.
It's not difficult to imagine Turing mentally picturing each of these people replaced with a machine that performed some simple task (without error). From there it's not a big leap to picturing himself as a "super machine", handing out computational orders to these drone machines according to a list of instructions - his color coding cheat sheet.
There's a lot to be said for immersion in the problem. We've all had the experience of working feverishly for days on a knotty problem only to have the solution spring, full-blown, into our mind during the morning shower. Turing was about as immersed in his problem as one could get.
Before coming to Bletchley, Turing had already invented the concept of the "Turing machine", which defined the absolute minimum computation needed to solve a host of problems. That already in his mind and his immersion in what constituted a crude, real world Turing machine would certainly have helped point the way to a mechanism to replace the people.
Also, he wasn't unaware of the immediacy of the problem he was working. Decoding an "attack at dawn" message two days after it was written is pretty useless. Certainly he would appreciate that a mechanized solution would be faster and the payoff of "fast" was supply ships actually reaching Britain.
Please note that I'm not suggesting that I actually know how Turing's mind worked. It's just that, with my own personal experience with immersion, I can make what I think is a fairly logical extrapolation.
Immersion isn't everything. While Einstein claimed that Special Relativity came from his imagining "what it would be like to ride on a beam of light", most of his breakthroughs involved concepts where immersion was clearly impossible. All of which makes what he did all the more incredible. Nevertheless, Einstein did immerse himself in a problem in a sense. His favorite trick was to perform what he called Gedankenexperimenten (thought experiments) in which he imagined things like what would happen if we bent space and made time a relative parameter.