I'm quite central so make the most of the royal parks but want some new scenery
Hi summer74, another Londoner here and totally understand what you say!
I have worked out though a few cheap ways of getting to the hills by combining cheap train tickets (offpeak or advance, obviously) and cheap rental cars from not so obvious places. For instances, you can get to the white peak area of the Peak District by taking a train to Stoke on Trent, which is about 45mins drive and usually has plenty of cheap advance tickets, and renting a car for the weekend from one of the dealers nearby the station, which costs about 30 pounds for 2 days.
Otherwise, an advance ticket to Edale will bring you right in the middle of some very good walking, and the same can be said of Penrith or Windermere. Advance tickets are sold 12 weeks in advance so it takes some planning to ensure you get the cheapest ticket but may be worth it.
The trick is to either travel offpeak or buy advance tickets.