Thanks, everyone, for your comments!
I admit that I haven't had a chance to do a lot of research into some of these places, so I don't have a great sense of everything that the locations have to offer. Below are some responses to the above.
I would say that we generally enjoy things that are cultural / art, architecture that's beautiful / unique, good food, walkable towns especially with lots of shops / markets / restaurants. The main thing is to see and eat stuff that we don't get in the US.
Regarding Bath, I think the long train ride is ok, especially since it seems like a great town to visit. Our kids like the train. And a year ago, we did a day trip from Florence to Ravenna, which entailed an equally long train ride and they did fine.
I know both Oxford and Cambridge are college towns. I was considering both because they both seem to get good reviews here and people don't seem to agree which is better, though I can't articulate the difference between them. Is there one that fits better given our interests above? Would it be a waste of time to see both?
I did pick Canterbury because of the cathedral. I don't know what else is there, but hope there'll be a nice walkable town / markets / shops to see as well. I personally like visiting religious sites and always make an effort to see major churches, temples, mosques, shrines, etc. in every country I visit. On this trip, we will be touring Westminster Abbey, St. Paul's Cathedral, York Minster, St. Giles in Edinburgh, and I'm sure others. I guess I don't mind seeing Canterbury Cathedral, especially if it has an attractive town that we can walk? But I'm not tied to it either.
As to Windsor vs Hampton Court, it was a difficult choice. I sort of picked Windsor because it looks more grand on its website and I think the family might be wowed by it (especially the senior, who has not travelled abroad extensively - she can tell her friends that she visited where the King lives). There's also a nice(?) town, so it's not just visiting the castle. The downside is that you don't get to see much of the castle due to limitations. I agree that Hampton Court seems more kid friendly, you get to see more of it, but it doesn't sound like there's any town there? Also, I guess as an American, I'm less intrigued by the history of the palace about Henry VIII and his various wives that he killed (this is what the website advertised). It might be a fun trip, but I'm not sure I can articulate "why" I should tour Hampton Court.
Is Holyrood Palace not worth visiting? It doesn't look as grand compared to Windsor on the website, and like Hampton Court, I'm not all that interested in the life of Mary, Queen of Scots. But I thought we might visit it because it's a royal palace that's actually in use and there's probably great art and architecture? We are definitely doing Edinburgh Castle and Stirling Castle, which I understand are both good but different.
I didn't think of Dover and can look into it.
I will add that I did consider Stratford-upon-Avon, but decided to cross it off because the kids and the senior won't be interested in Shakespeare (at least not enough for a day trip). I also crossed off the Cotswolds because while we would enjoy the scenic small towns, they're logistically difficult with public transit and I don't want to rent a car and drive.