A pretty quick work trip has been planned. I will fly into Heathrow Saturday morning. I'll have until Sunday evening to get to Warwick. I'll be in Warwick for 5 days before flying back out the next Saturday.

I'm looking for any general advice but also if there is anything specific to the following:

• I'm going to book my own hotel in London the Saturday night I fly in. Saturday and Sunday are essentially my "tourist" days. Where is good to stay? Not too concerned with price.
• I'm taking a train to Warwick and mostly have that figured out but is there an app or pass that I should add to my Apple wallet for transit around London?
• I like museums of all types. Are there any in particular I should check out?
• Any classic pubs or restaurants I should try to get to?
• Once the week starts I don't think I'll have much time to do touristy stuff and I won't have a car. Any recommendations on things around Warwick/Birmingham that I can get to some evenings via train or bus?

Looking for ideas - we're two adults in our mid-30's, no kids. We'll be leaving a wedding on a Monday in Ipswich but our flight home from London isn't until Friday. We'll be spending a week in London beforehand so we'd like to explore the countryside.

The only thing we've (sort of) landed on is seeing/staying in the Cotswolds, and touring some castles (holy crap there's a ton to chose from). We're not big drinkers/partiers but I'd like to be able to go hang out in a real English pub and stay at a Lord-of-the-Rings kind of inn (I already found the inn that the Prancing Pony is based on - thought not sure if we want to stay there).

Other random thoughts. Unfortunately our schedule doesn't seem like it'll line up with any Premier League games. Not planning on renting a car but can if we absolutely must. Would kind of like to stay in one spot since it's only 3 nights.