London and Cotswold stays before and after Norwegian Cruise in early October 2025

I will bow to the wisdom of our UK transport experts, but I believe the direct trains to Southampton leave from Waterloo, so staying near there would be a choice. Trains requiring a change leave from Victoria and Paddington stations. Advance tickets can save money over an Anytime Day ticket.

If you want to stay in the Cotswolds, but not drive, then your choices shrink. If you want to travel by train, and not stay in a city, they shrink further. You won't easily find a small Cotswolds village (with amenities) on a train line. But Moreton-in-Marsh, a town, is likely the option more frequently mentioned. You would probably need to connect at least once from Southampton to MiM, and again from MiM to London.

Yes, the direct fast trains to Southampton leave from Waterloo. As long as you are willing to be tied to a specific timed train then the sooner you buy the better.
However it is unlikely you would pay full price buying on the day as there are almost always walk up Advance fares for specific trains, albeit at a higher price than buying today.
From Southampton there is an hourly train to Oxford, change there for Moreton in Marsh.
From Moreton in Marsh there is an hourly direct train back to London Paddington.

If you really want to see the Cotswolds, do not be discouraged by the lack of trains. We found that with a little advance planning, we had a very enjoyable, 5 night stay in the Cotswolds using a combination of buses and private drivers.

And if you are near Moreton in Marsh there is a great tour company that picks up there. I took a Go Cotswolds tour when I was there in 2022 and really had a great day. It was a large van that zipped around to many places of interest.

I had this page open to respond to another post, so here is the general information on train travel in UK, with a section on how to buy tickets and strategies for getting good ticket prices:
https://www.seat61.com/train-travel-in-britain.htm#Cheap%20train%20tickets
I am a major AirBnB user and former host, BUT for "a couple of nights" in a big city like London, traveling on my own, I would probably stay in a hotel for a variety of reasons (none of which is safety, BTW. London is a very safe city for tourists, to my knowledge).
I think the other commenter's suggestion of staying near Waterloo makes sense. Looking at the map, there are a couple of hotels right on the river with crazy views. I can't vouch for any of these, but here's one: https://www.seacontainerslondon.com/?utm_source=google&utm_medium=local&utm_campaign=hotel-seacontainerslondon
You could do a search on this site for hotels in Waterloo area.
A quick search in that area on AirBnB showed substantially higher prices in the Waterloo area than for hotels.

We took a wonderful cruise of Norway all the way to the North Cape in 2019. It was with Royal Caribbean and wonderful. Our cruise was in July.

If your cruise is in October, you will probably be limited on what excursions are available due to snow.

Now, to respond to your questions.
We did 6 nights in Chipping Campden at the Volunteer Inn (In the Cotswolds) it was great, love the Inn. However, we rented a car and my wife constantly warned me to "stay on the left" when driving. After a few hours, you get used to driving on the left.
We used this as a base to visit the Cotswolds for two days and also visit Oxford, Blenheim Palace and Stratford Upon Avon.

The rail connections to the Cotswolds are very poor as others have explained. I recommend renting a car, British drivers are very polite, but the traffic around the London area is terrible.
We also took a transatlantic cruise to the USA from Southhampton in 2017 and were able to drop off our car near the port.

Other options besides the Cotswolds are Bath, Winchester, Stonehenge, Salisbury on the west side of London and Canterbury and Cambridge on the east side.