Moving from the south, to the north of England?

Barnard Castle is a good choice. The towns that you mention in Additional Details 3) are quite a way from the Lake District. I live east of Leeds, and for me Windermere (for example) is more than 2 hours' drive away. Skipton would be OK, I think. It's about halfway between the eastern edge of the Lake District and Leeds. If you live in Leeds, and drive to the Lake District, you have to p through several built-up areas before you even get as far as Ilkley, where the open road begins. That can seem like a bit of a marathon. So, Barnard Castle or Skipton. Put them on your list! Best of luck.