We just returned from England last night and I wanted to share our experience, we did travel from London to Bath as well during our trip. We purchased almost all of our train tickets in advance through LNER and found using their app to show tickets worked well. We also bought a friends and family rail card, used the app (which you do have to show when you show your tickets) and it saved us a bit of money traveling as 3 adults - my daughter is 17 so counted as adult - and 1 child.
Having said that, we ended up traveling during the strikes and I ended up refunding and rebooking tickets while I was standing in train stations. We bought in advance because I am a high anxiety traveler, haha! I really LOVE a plan! But honestly, my husband and I both agreed in the future, unless it was a very specialized route, we wouldn't buy advance tickets again. We traveled London Paddington to Bath Spa and there were plenty of trains to choose from and so easy to just buy right there at the kiosks. We still bought through LNER because they were so wonderful about refunding our tickets due to the strikes, we found reserving seats through LNER much easier, and their app was very good and stable in every location we were in. If I were doing it again, in my opinion, I would just buy when I got there. But if buying in advance, I found LNER was the best way for us to go!