Yes it will be very busy.
But, and it's a big but, people make their own problems by a) driving here and around the district; b) all staying in the same few honeyspots like Keswick.
I am sorry but RS is one of many culprits in that regard with his you must stay in Keswick mantra. Even if you require Keswick think Threlkeld, or Braithwaite/Thornthwaite, or SJITV (other places in a 3 mile radius), or as I say Cockermouth (8 miles away).
We already had good public transit, which has been hugely improved this year. For Pete's sake in your own interests and everyone else's please use it.
Us locals get very irate when we can't get round the area on our everyday business because of the lines of tourist cars at entirely predictable pinch point locations. But then we also know the avoidance routes as well!!
All these unnecessary cars is why the central area (Windermere to Grasmere, with lines of traffic back to Ings trying to get through Windermere) gets so congested and why car parking gets so tight or almost impossible at such locations.
Locals and the police are beyond exhausted with the parking issues at places like Catbells, Buttermere and Elterwater, to name but a few.
Try staying down in the Furness area instead (most of which is all but undiscovered), or Kendal, or Penrith, or out on the West Coast here. Cockermouth would make a great base. We even have Premier Inns at Cockermouth, Workington and Whitehaven as well. Both Workington and Cockermouth work very well (Whitehaven just a bit less so), and we also have tourist attractions out on the West Coast as well, our own history to explore, our own great scenery and walks.
Back o'Skiddaw does really need a car, but if you stay up in that area having a car won't be the problem it is in the Central area.