Where to go outside London

Does it have to be The Cotswolds? So many nicer, less crowded places.

Suffolk? Check out Orford and Orford Ness, Southwold.
North Norfolk coast? Cley, Blakeney,
Dorset? Abbotsbury, Lyme Regis
New Forest
Rutland (the same stone outcrop The Cotswolds is made of. Virtually no tourists.
Kent? Whitstable, Rochester

if you go to Cambridge ( an easy train or National Express bus ride) there are a variety of options to consider and Cambridge itself is a great place to wander and explore. You can divide your time as to what most interests you. Explore nearby Ely or Bury St. Edmunds by train or bus. Ely's cathedral is majestic, perched on a hilltop, visible across the fens. Explore the historic Ely town center near the cathedral and walk down to the riverfront. Bury St Edmunds, is another pretty town, easily accessible. It has beautiful gardens at the Abbey ruins and is a market town so always interesting on market day. Nearby is Ickworth, a pretty National Trust Property but you may need to go there by taxi. Not too far from Bury St. Edmonds ( but you would have to go by taxi or check out a bus schedule) is the lovely little town of Lavenham, filled with its crooked timbered houses. It is perfect for wandering around. Very photogenic and its quirky homes have been featured in several movies, including one of the Harry Potter pics. We used to live in Suffolk and Lavenham was always our favorite place to take guests. Right on the edge of Cambridge is Anglesey Abbey, Gardens and Lode Mill. Very pretty. Not far from Cambridge is another pretty and interesting National Trust property, Wimpole Estate. if you like WWII history and aviation there is a huge Imperial War Museum at Duxford Airfield. Also nearby is the Cambridge American Cemetery and Memorial. Lots of choices and places to explore. Hope this helps.

The conventional wisdom is that you have to drive to tour the Cotswolds independently which as far as I know comes from the RS guidebooks.
That is not strictly true. There is a bus company called Pulhams- https://www.pulhamscoaches.com/bus/ -who operate a perfectly reasonable service in the Cotswolds.
Yes it is far more convenient in all kinds of ways to drive yourself, but transit is not as impossible as everyone makes out. You go to Moreton in Marsh by train (hourly service from London Paddington) and then bus from there.
The big questions for you are-1) how used/experienced are you to using transit, 2) are you confident in using transit in the UK (or even in Denver) and when it is not along every few minutes as in a big city. 3) Do the dis-benefits of transit outweigh your wife's wish for you not to drive on the other side of the road.
I can't answer those questions for you and certainly won't insult someone who I don't know.
All I am doing here is answering your question that there is another option, which was your question.
I am sure others will be along to convince you that the transit option is not viable and to encourage you to drive.
This post opens that debate.
The other part of the question about anywhere else, villages wise from Oxford, is that you can easily do some very nice villages and towns along the River Thames towards Reading (good bus and train options) or go east from Oxford towards the Chilterns (again bus and rail options). Both of these are often overlooked areas as against the Cotswolds.
Oh, and Blenheim Palace (a very easy trip from Oxford) should ideally be on your agenda.

or do 2 daytrips, in different directions. I did Bath in a day, as did the Man in Seat 61, it's simple.

Thanks for the excellent responses so far, please continue, I'm definitely paying attention!

I agree that many USA tourists have little or no experience/tolerance for public transit. We've used it extensively on our seven visits to the UK and Europe, and on many trips to major cities in the US. It's a valid concern, but I think we're good on this point.

As to alternatives to the Cotswolds we're flexible, just looking for convenient access to some village-like places that we can manage within our limited schedule.

Also, consider Windsor Castle, Canterbury, Salisbury and Stonehenge for day trips.

I suggest spending a couple on nights in York, there is a lot to see there.

As for the Cotswolds, you need a rental car, rent it in Oxford, it is a short drive to Chipping Campden.
We stayed there at the Volunteer Inn, which has the best pub in town and a great Indian Restaurant. Also, spacious rooms for reasonable price.

Rent an automatic transmission car for the Cotswolds, driving there will not be stressful. Have your wife can help you navigation with a phone loaded with Great Britain maps. If you don't have such a phone, make sure the car has a nav system.

We did four weeks in Wales and England in 2017 when I was 69 years old. No big deal, just be careful and don't speed.

1.) The Richmond neighborhood. Over the years of visiting my favorite city on the planet I’ve learned to stay outside the city proper. Do this because I’ve seen the major sites and enjoy exploring outlying neighborhoods. Places like Richmond, Ealing, Chiswick, Highgate.

Most recently in Richmond I used busses and the tube from The Quadrant to venture about other neighborhoods.

In 2021 stayed at the Premier Inn and went to see A Christmas Carol Ghost Story at Alexander Palace. Bit of a jaunt via the tube and a bus but loved it. Was bemused when waiting for the tube train at Crouch End and noticed actor Nicholas Farrell who was one of the stars of the show doing the same. Remember thinking “ dang that’s a long commute.”

Lots of a pleasant cafes, restaurants, shops, pubs in Richmond. Also the Richmond Theatre off the Richmond Green. There’s also the gorgeous Richmond Park to meander about. I’d enter the park via the Kings Gate entrance next to the East Sheen cemetary.

Stayed at the Premier Inn on Lower Mortlake. A venerable non gastro pub is adjacent to it. Good pint of Guinness and liked the Thai food.

2.) Train from London to Whitsable. Day trip or stay overnight.

3.) Train from London to Bletchley to visit Bletchley Park.

4.) Stay in Cambridge and bus to St Ives.

5.) Stay in London for few days. Train from Paddington to Morten In The Marsh. ( 1.5 hour) Stay over night there and bus to Stow-on-the-World. (25 minutes)

6.) Stay in Oxford and bus to Blenheim Palace.

There is a good reason why many people go to the Cotswolds as it is a very pretty part of the country. I am heading there myself this year and taking some American friends with me staying in Bath and taking a MadMax tour.
I grew up around Cambridge and the Suffolk/Essex border. Cambridge is easy by train and has much to offer as well as the nearby Ely Cathedral (I visited last summer) - It is indeed Majestical. I do not care so much for Bury St Edmunds, but Lavenham is lovely - but a car would really be needed for that. I just think you would find the buses slow and you would end up standing around a lot. There are other parts of England that I would probably tell you to go to such as Yorkshire or the Lake District or Even Kent. I visited Kent last year also and stayed at Leeds Castle, Nr. Maidstone (surprisingly reasonable) and visited Canterbury and Rye. Kent is a lovely county. I must admit it is hard to pick an area, in England.

As for driving, I am 56 years of age, and I passed my driver's test in England at the age of 18. I will say that while I find driving on the left with a manual transmission, not too bad, I get quite anxious in heavy traffic, in roundabouts that have several lanes, and with all the fast and aggressive drivers on the road. It's not like driving across the US but If you drive in the cities in the US you may be OK in England.

But the more I read it seems the only way to visit the Cotswolds is to take a tour or rent a car - neither of which appeals to us. It seems there's no convenient bus service from one village to another. Does anyone have any ideas I may have overlooked?

If you want to get into the villages you are correct.

Regarding the Pulhams reference, I grew up using that company, and they are old and generally well respected. But they link towns much more comprehensively than villages. Moreton in March, Stow on the Wold, Cheltenham. Villages are much more times for market days or commuting or school runs. Check very carefully before relying on them.

The £2 single bus journey cap has been extended throughout the country through June now.

The key question to consider whether you should drive in England is have you had experience of what for you is the “wrong” side of the road. If not, I would suggest that age 74 is probably not the best time to start. Many of us in the UK get the chance to do the opposite on most holidays outside the country. I’m 70 and I’ve been doing it off and on for 45 years. I’m not sure I would want to be doing it for the first time now, although of course I like you am an excellent driver!
Going to the Cotswolds the main thing you will see is a lot of your fellow US citizens. As others regularly point out it is by no means the best rural area to explore. To other suggestions I would add the Welsh border counties (Herefordshire, Shropshire and Powys) or further afield, the Lake District, Peak District, Yorkshire Dales and North York Moors. The latter four are all National Parks, unlike the Cotswolds. And they can all be reached by a 2/3 hour train ride from London.

Whereas I, from the UK, have never driven on the other side of the road.
I am in the middle of planning a visit to WA but at my age am distinctly unenthused about driving there.
My first night of the trip (at least in WA), having come by ferry from Vancouver BC, would be in Port Townsend for one of the non-negotiable items in my itinerary.
I thought there was very little other useful transit on the Olympic, so was scheduled on Day 2 to continue on the more interesting of the two possible WSF routes to Seattle via Whidbey Island.
Earlier in the week on another thread someone commented about using city bus from Bellingham to Seattle (which I already knew was possible, and fun) and mused about the possibility of going on by citybus to Portland, Oregon. That spurred me into thinking, is that possible.
And suddenly from what was almost an afterthought of a comment I found that the Olympic peninsula has an incredibly good transit system (and amazingly cheap) so a lot of places which I thought were inaccessible without a car (even Cape Flattery) suddenly are, and in what for me at least is a way which is much more interesting, even mildly adventurous.
That one, almost incidental, post has just added days on to the itinerary!
So it is amazing where posts on here can take you.