I’d go for Kesick or Ambleside to start with.
The National Trust campsites we’ve stayed at are great. They don’t allow cars on the camping ground which means you get much better views and a lot less slamming of car doors at night.
Low Wray and Great Langdale are both good bases for riding around Ambleside. You can easily do a few great rides straight out of the campsite at either (and pretty much all of the major Ambleside riding if you’re prepared to put some miles in). The area between Windermere and Coniston is packed with good trails – most of the bridleways are interesting but pick up a route guide from Biketreks. Langdale – Grassmere – Loughrigg Fell is great. I think we’ve ridden the Garburn Pass via Troutbeck from Low Wray as well in the past.
For Keswick either of the campsites at Castlerigg are fine. You can ride the Borrowdale Bash and Skiddaw rides from there (you finish both with a climb to the top of the hill but you can do it on road if you’re tired).
National Trust as Wasdale Head was also good but riding there still unfinished. One of the few times i’ve ever turned back from a ride – we were heading up to Black Sail Pass when a big storm came in. Enough to flood the campsite so was probably the right move.