I love Wells and have been a couple of times. DO take one of the docent tours of the Wells Cathedral. IF you have time go in to the BIshop's Palace too. It's small but interesting and the gardens are pretty neat. You can get up onto some of the wall there if that's doable for your Mom.
Many enjoy Glastonbury. I'm glad I went but I was not totally taken with the ruin and the whole King Arthur and Joseph of Arimathea legends. I'd probably visit the abbey ruins then walk thru town a little. There are way too many New Age shops for my taste. In the Abbey Visitor Center there is a nice but small museum. I enjoyed the medieval kitchen and herb garden.
I'm not sure how much driving you want to do but Cerne Abbas is not too far away where you can see the hill figure of the Cerne Abbas Giant. The village of Cerne Abbas is cute as well.
https://en.wikipedia.org/wiki/Cerne_Abbas_Giant
I love Bath but I really love the Georgian/Regency era in addition to the Roman era sights. I've visited a number of times and stayed up to 5 nights, lol!
I'd vote for Salisbury if your Mom hasn't been there. That cathedral is just amazing to me and one of my favorites. They have some interesting art installations in the nave at times. Last May there was an installation of origami doves floating down the center aisle that was very powerful imagery in view of the horrible poisonings/killings of this spring.
June is a great time for gardens but the ones I've been to have been further out in Cornwall so too far to go to. I'm sure someone will come up with a stately home with a lovely garden in the area!
editing to add: Here's a link to the National Trust properties. Click on the 16 and that will put you in your local area.
https://www.nationaltrust.org.uk/search?view=map