Just to add to what has already been said. You can avoid the railway track outside Keswick by going along the road past Castlerigg Stone Circle. I have done the railway track on a hybrid, but can't remember if it would be OK on a road bike. Also I think part of it may have washed away in the recent floods. Don't try and follow the off road section from the top of Whinlatter, just carry straight on along the road.
If you are going to Tynemouth, you can't take bikes on the metro, you will have to cycle back into Newcastle. If you are getting back to Whitehaven by train the Carlisle to Whitehaven train only has three bike spaces, which can't be booked in advance. You may be able to take more bikes if you get a friendly conductor.
We did Whitehaven to Penrith on day one. Then to Allenheads on day two. In retrospect we would have been better carrying onto Stanhope, as day two was quite short. Crawley Bank is undoubtedly the toughest climb on the route!
Edit: If I was doing it in two days, I would go all the way to Allenheads on day one.