I did this hike about 9 years ago in two weeks time. This is where learned about going light-weight, especially in the low-hiker shoe department. St Bees is a tiny town, and the train to it is more like a commuter train with personality. Just make sure the person driving the train doesn't pass by your stop. I flew into Manchester and then took multiple trains.
I did do it with a group, and glad I did because nothing is marked. There's sections of road walking ( 24 miles of it in one day). The only place to stay up in the moors is a tavern. You can book ahead. I went thru a company called Sherpa based in the UK. They can reserve the spots for you to stay. A GPS might be very helpful.
I loved the entire hike. We had only a few rainy days, mostly in the Lakes District. Great hiking...hard on the knees going down the fells.
The end at Robin Hoods Bay has a bus system that will take you to the train station which goes directly to the airport. Things may have changed since then.
There really isn't any place to camp. The pubs have rooms, as well as B&B's in most towns. Some towns are very tiny, so plan well. Nothing like walking into a warm pub, good food and rest after a cold damp day of hiking.
One of my favorite B&B's had a huge rock in the middle of the living room. They built the house around it.