Oxford? Easy to reach from London, historic colleges and museums and since your kids are university age they may enjoy seeing Oxford. It's a nice day trip.
York is another doable day trip from London. Great cathedral town surrounded by old medieval walls. Fabulous cathedral and historic city center. There's also Jorvik viking center, which your children may enjoy.
I'll agree with another poster that providing recommendations is stabbing in the dark a bit without knowing what you really like to do. Are you interested in castles and historic stately homes (are your kids budding architects?). There's Hampton Court outside London, and Knole in Sevenoaks, also outside London. Knole is one of the most ancient houses in Britain and well worth a visit if you're interested in Elizabethan and Jacobean history and decorative arts. It's quite magical. Both are easily reachable via public transportation from central London. But your kids may be bored if they only care for video games or hip urban markets.
Bibury is a lovely village but it's one of those places that once you get there and walked around for half an hour, you've seen it and there's bugger all to do. It's best used as a base for exploring the Cotswold, rather than a destination in itself. If you want to see a lovely English town then consider Salisbury, which has one of the most beautiful cathedrals in England, and is surrounded by the famous Salisbury cathedral close. The town itself is quite lovely too. You could combine a visit to Salisbury with seeing Wilton House, one of the great stately homes of Britain, which is just outside the town.
I know Broadway market well enough. And I won't recommend going there. Why? It's geared to London residents, not visitors. And residents who live in the general vicinity of Broadway. And it's not particularly big, just a few blocks long. You'd probably spend more time getting to the market then at the market itself and you're not going to see anything at Broadway that you can't find in the other markets in a more central location. I wouldn't advise going to Broadway over, say Borough Market or Camden as the latter two are much more centralized and bigger. Yes, they're more touristy but you're a tourist.