Derby I haven't lived in but know people who have. It's an old manufacturing town and hasn't much to offer graduates. Or anyone really. The Peak District is great, and if you can live out that way and commute in then do it. But again, you won't have similar people for local friends.
Saying the peak district is good for young people is like saying there's a great lake near Detroit, it's not exactly what they're after.
It's also one of the wealthiest areas outside of London. But house prices in the really nice parts of Solihull are also high.
When I live in London I didn't drive, which was kinda nice but also meant I've only been out of city like once a year.
Sitting in traffic sucks of course, but driving rurally opens so much.
As for weekends - driving and hiking I guess?
For example I lived not far from Putney. Putney to Windsor & Eton Riverside takes 39 minutes and costs £6.90.
https://www.openstreetmap.org/#map=11/51.5330/-0.1146&layers...
Is it as glitzy as London. No. But saying there is "no culture" is just absolutely asinine.
I live on the outskirts of the peak district. I can walk/cycle less than 30 minutes out of town and be walking along the old canals, through old villages and get amazing views of the countryside.