As a former Lake District resident, it's really hard to say. When there is snowfall around there, it can be quite variable and unpredictable as to where it settles and how much disruption it causes. I would say you should be fine. One good thing about such a rural area is that often local farmers and other residents will get out and clear the roads around them, so you'll often find minor roads are quite passable even when there's a lot of snowfall.
I would say you should make sure you have travel insurance, so that if it does snow badly, you'll be able to get your money back.
To book a break in the Lakes in February next year? Have found the perfect place to surprise DH with, and was thinking of going in March or April. However.... the price has been significantly reduced in February so am tempted to go then. Its fairly remote - near Wast Water - but isnt particularly high at about 200ft.
Is there likely to be lots of snow lying then? We have autosocks for the car so some snow wouldnt be a problem but I wonder how likely it is to be cut off completely? It would be really disappointing to book but not be able to go