I moved to Huddersfield 1.5 years ago from Norfolk, my take on the local cities.
Leeds – nice city, good bars & restaurants etc with plenty going on. Locationally its ok, but the peaks i.e. ladybower is a good drive. Dales is closer but tbh ive yet to find any half decent riding compared to Pennines or Peaks.
Sheffield – compared to Leeds its not as good, wouldnt want to live in Sheffield but it is easy to get to Peaks, however anywhere else is longer drive i.e. dales, pennines, wales etc.
Manchester – loacationally best placed for getting to the hotspots around the country, if you live stalybridge or stockport way getting to peaks is easy and hayfield is really close with some top riding. Manchester is a great night out too, loads of decent places to eat and drink and again like leeds loads going on but it is ridiculously busy and takes an eternity to drive out of.
Personally if it was me, I’d stay in Huddersfield. Dead easy to get to leeds and manchester on the regular transpennines trains for the bars and restaurants. Has the much less touristy north peaks on the doorstep and its not half as busy as the cities.