Glastonbury, Wells in a day from London

Sorry, but this isn’t practical as a day trip by public transport. The closest rail station is at Castle Cary (trains from London Paddington) about 12 miles from Wells. Wells has a great cathedral, however, Glastonbury has little to recommend it, other than walking up the Tor unless you want to visit overpriced vegan cafes (the most expensive slice of cake I have ever had in the UK and certainly not the best) and to buy some joss sticks.

Car hire would be your best option.

You can take a train from Castle Cary to Salisbury (for Stonehenge) which takes 80 minutes, changing at Westbury. You would then need to take the bus to Stonehenge. It’s less hassle just taking the train there from London.

The best way (that I know of) to get to Glastonbury and Wells by public transportation is by bus from Bristol. Information about the bus route is here. From Bristol Temple Meads train station to Wells is about an hour, then it's another 15-20 minutes (stay on the same bus) from Wells to Glastonbury. (There's also a bus to Wells from Bath, but it takes longer, so it's not worth considering unless you're already in Bath.) The two are very doable as a not-terribly-taxing combined day trip from Bristol. I suppose you could add in the round-trip train from London - that would make for a rather long day, but not absurdly so.

If you wanted to add in Stonehenge, I would suggest: Stay overnight in Bristol or Bath (when you're done in Glastonbury and Wells), then then next day take the train to Salisbury, then bus to Stonehenge, then bus back to Salisbury, then train back to London. There may be more efficient ways of doing it, but that should work.

I disagree that Glastonbury has little to recommend it. The ruins and grounds of Glastonbury Abbey are beautiful (and I assume that's the main reason you're going there).

No, you will not be able to fit Wells, Glastonbury, and Stonehenge in 1 travel day from London, especially without a car.

Unlike other opinions, I think Glastonbury is well worth the visit. Yes, it does have that "hippy" vibe people speak of, but the history is magnificent. I stayed the night in Glastonbury last June, at the George & Pilgrim Inn ( medieval building that dates back to the 1400's). Glastonbury, in addition to the tor, has the Glastonbury Abbey exhibit that I thought was totally worth it.

Wells also provides wonderful charm and medieval ambiance. I am returning again this year and staying at The Crown at Wells inn, which also dates back to the 1400's. Wells Cathedral is the best I have seen and Bishop Palace Gardens will offer a nice stroll with medieval charm.

You can fit Wells and Glastonbury in one day if you manage your time right, but be sure not to rush as well. It may be a bit more difficult for you since you do not have a car and your valuable time may be affected due to this.

I took the bus from Heathrow to Bath, then rented a car to get to Wells and Glastonbury. I would probably suggest the same.