Jealous! The Cotswolds at Christmas is near the top of my own bucket list. if I could just get my siblings to stop shooting it down, I'd have gone, but they don't like to travel unfortunately.
For the Cotswolds, I'd look at Broadway, Stow-on-the-Wold, or Cheltenham, depending on what kind of experience you want. Broadway would be my first choice if you want the idyllic Cotswolds experience, it has a great high street with pubs and restaurants. Stow is a little more traditional and centrally located, so I'd choose it if you want to explore several of the surrounding villages. Cheltenham is a larger town on the edge of the Cotswolds, with more hotels, restaurants, shopping and Christmas activities. I'd choose Cheltenham if you want the convenience of a hotel and more going on around you, especially if you're there over Christmas itself.
I'd give yourselves 3-4 nights there rather than trying to squeeze it into a day trip (plus lots of places have minimums anyway). In December, I think the appeal is as much about enjoying the village, having a pub dinner, and slowing down as it is about sightseeing. If you're actually there over Christmas, I'd probably lean toward a hotel/inn with a Christmas package rather than a rental so you don't have to worry about meals and logistics.
In London, I recommend Covent Garden or West End for Christmas time stays. Mayfair is also good but tends to be a bit pricier.
I've booked a few England holiday trips like this for people, so feel free to reach out if you have any questions or are looking for help putting it together.
ETA: Another completely different option: if you're looking for a really memorable Christmas experience, look at the British Pullman. They run festive day trips from London to places like Blenheim Palace and Bath on the luxury Pullman train. It's definitely a splurge, but could be a really cool alternative to adding another hotel stop.
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