I had the privilege of knowing the late Ann Welch very well, who had been a wartime ATA delivery pilot. She described to me that she received a basic "class" training (training aircraft, fighters, bombers...), and then was given a set of "ferry pilots notes" which contained the basic information to operate the aircraft under the conditions they required.
I have no doubt this was a huge exaggeration (I know what a safety obsessive she was throughout her life), but she did say to me once that having arrived at an an airfield, she'd be looking up how to start the engine as she was driven to the aircraft, how to take-off whilst running up, and how to land during the cruise.
But it does make a point - any TP who is routinely switching aircraft types now will be doing something similarish to what Ann was doing back there - relying upon clear sets of notes carried in the air that give them the information they need to safely operate the aircraft - above and beyond (and probably including) what they've memorised.
I don't know for certain, because I'm much too young and whilst I've met several wartime TPs I've never had that conversation. But, I suspect that a TP of the period would, not dissimilarly to the ATA pilots, have been very reliant upon type-specific notes and self-brief. The big difference is that the TP would most likely have prepaired much of the notes themselves, as they mostly would now.
Now having given a chunk of marginally informed rambling, it does occur to me that if you can get in touch with Eric Brown, he'd be the man to answer this question with considerable authority.
G