I live in Newcastle (on the other side of the coast to the lakes) and like you I do not drive, and I find getting over to the lakes on my own can be a problem.
You can get a train to Windermere, which is in the southern lakes. Another possibility is Ravenglass on the west Cumbrian coast. There is a tourist train from there that takes you into the mountains. If you get a train to Penrith you should be able to take a taxi or bus into the lakes (the town of Keswick is an excellent base).
Any of these options will probably take the best part of a full day of travelling from London, so your walking time in the lakes would be limited. However, if you can get up there then you will find it to be a very special place (even if it is raining)