People like Turing were certainly familiar with Babbage. Babbage was well-renowned and a Royal Fellow. Whether on a mechanical or electrical computer, computer science remains the same, just like astronomy is not determined by what telescope one uses. Babbage really laid foundations for Computer Science paving a path for Turing to use.

The first lesson is that computational complexity theory is really, really, really not about computers. Computers play the same role in complexity that clocks, trains, and elevators play in relativity. They’re a great way to illustrate the point, they were probably essential for discovering the point, but they’re not the point. Scott Aaronson