Nothing wrong with a bit of driving, i’ve done Birmingham to Scotland for a long weekend enough times, and a week in the alps is a day each way with 3hrs driving each day sometimes more.
I’d personally bypass the lakes and head further down, you could do the lakes in a long weekend easily enough so although the lakes and yorkshire are some of england’s best riding i’d crack on past personally.
Start in North wales, South wales, devon/cornwall, across the south coast then back up to the midlands and do a weekend in the peak district on the way home.
Plenty of guidebooks for those areas, loads of trail centres & gps online options. Grab a days uplift at Stiniog or Bike park wales if you can. Penmachno, Coed-y-brenin, Llandegla, Cwm Carn and Afan are the obvious welsh ones but check out the black mountains, berwyn mountains, Machynlleth as well. I could spend a week just in wales 😆
http://www.theedgecycleworks.com/mtbroutes/
Sleeper train to london for a weekend another time…