If I remember well, an Allison J-33 engine or RR Nene had something like 30 seconds spool-up time from idle to max power...?
I don't know about the Allison, but I flew the Nene powered single seat Vampires and spool up time on a touch and go was quite fast -around 5-8 seconds from idle to full power. The dual Vampire Mk 33 with a Goblin wasn't too bad either although if you pushed open the throttle too fast the engine made loud rumbling noises.
Regarding the observations on reluctance of pilots (SOP maybe) to switch off the flight directors and automatics. From listening to countless stories of my former simulator students now happily flying jets as first officers, I was surprised to hear that most of them would love to be able to hand fly raw data but their captains are often reluctant to let them do this. All sorts of reasons are offered for this reluctance but it boils down to the fact the captains were themselves rusty at real handling (as against letting the automatics do the lot).
There are plenty of willing young first officers out there who would like to dare to switch of the FD in a climb or descent - but the left seat bloke won't allow them the pleasure. It's all about the "twitch factor."