If you decide to use the trains, note that Cross Country Trains go from Edinburgh to Bristol Temple Meads (15 minutes by another train to Bath) but CC go via Newcastle & York - which is a longer route by about an hour. The quickest & shortest route is to go with Avanti West Coast via Carlisle to Birmingham and then Cross Country Trains to Bristol - where you change to a GWR train for Bath.
Swapping Bath for York would be much easier as York is on the East Coast Mainline from Edinburgh to London King’s Cross.
Buying a through ticket from Edinburgh to Bath can be very expensive. It is best to buy in Advance (maybe 9 weeks out) for specific trains and split tickets works out cheaper. Test out these sites as they will tell you your options. Note that they take a commission on the amount saved. If you don’t want to pay that commission - you can book with them but each leg separately having first noted down the tickets to buy.
https://www.traintickets.com/?/
https://v3.trainsplit.com
Note that Easyjet charge for baggage in hold.