Other than our very first trip where we did a Frames Rickard's "Windsor, Bath and Stonehenge" trip, we have done the day trips on our own. Other than in Inverness Scotland, where we found some excellent tour companies that do small group (10-15 people) using small vans, we have found the big-bus trips to be mostly time on the bus. However, I have read on this forum that London Walks does some 'by train' day trips that are well guided and nothing like the big-bus trips.
We have found the rail system in England and Scotland to be easy to use and generally affordable- though we often used BritRail passes (this was when they were still a bit of a bargain) because we liked the flexibility of being able to pick our travel days based on weather, or mood, etc. The only 'day of' tickets we usually bought were for Windsor. If you can commit to a specific day and time, then you can get real discounts by buying your point-to-point tickets about 12 weeks prior to travel time. Or you can look into some of the special travel deals promoted by various rail companies. Things like senior discount cards, special group discounts (even 2 counts as a group for some), etc. If you do go to London, be sure to look for the special street light displays along Regent's street, and other main roads. And watch for the roasted chestnut street vendors!
Instead of Rome, you might want to look into Austria. We LOVED Salzburg in the winter- it's Christmas market is fabulous, and so is the Austrian winter food. Innsbruck is also nice in winter (and we don't even ski), as is Vienna- both have good to excellent Christmas markets. We even did a day trip by train to Budapest from Vienna one winter. I know its cold there in winter, but again, layers, layers, layers. It snowed one year and we have lovely pictures of Salzburg in the snow. (and it would be a real change from your typical winter break). Or look into Scotland. We went to Inverness and Edinburgh from London one winter- it snowed then , too. Edinburgh has a good Christmas market and plenty to see and do inside if the weather is a bit brisk.