I lived in London from 2015-2018 and made it my ~bimonthly treat to visit a new course that I had heard good things about (often from people on this website). I did all of them by train. Most are pretty easy. With some, you can walk from the train stations but with others, you'll need to take a cab. But there wasn't a course that I wanted to see that I couldn't. And I had the same experience as Scott--there's no issue with a guy walking through the gate with his clubs (or getting out of a taxi). But with several of these clubs--especially Rye, West Sussex, and Swinley Forest--I had made advanced arrangements and I think that was important. I don't think you could just show up and get on.
The toughest from both an access and logistics standpoint was Rye. It took a few phone calls, a few trains, and a bus to get there. But it isn't a London area course and unless you're spending several weeks in London, I wouldn't recommend it--there are too many great places that are easier to get to. If you want to do one longer journey, you might try Royal St. George's/Deal/Prince's.
Most courses will require a taxi from the train station. But many of the train stations are big and it's very easy to get one. Woking and Worplesdon are easily accessible from the Woking station, which is one of the biggest in the London suburbs and is one stop on the express train from Victoria. Same for the Addington, which is near the major East Croydon stop. Although there's a tram from the station that goes in the direction of the course, I'd just take a cab. St. George's Hill is easy--near Weybridge station, which is one of the closest to the city among the big courses and just a few stops from Victoria. Walton Heath and Royal Ashdown Forest are easy to get to from London Bridge station.
Two much longer cab rides from their respective stations are Hindhead and Hankley Common. Both because the courses aren't as good and they're more difficult than the above to get to, I'd recommend skipping them unless you plan to play more than 8 or 10 courses. West Sussex and Swinley Forest are also a bit further out, but are absolutely worth the extra effort.
If you don't mind going a bit outside of London, I'd highly recommend Stoneham near Southampton. To me, this course is on the level of Woking and is near the Southampton airport station, which is an express stop on one of the trains from Victoria, so it's only 4 stops and takes about as much time to get to as several of the London area courses.
There are also several courses where you can walk from the train station. You'll definitely walk from Sunningdale, which is just a few hundred yards up a hill from Sunningdale station. You can also walk to West Hill from Brookwood station (~10-15 min) and Liphook from Liphook station. The latter is probably at least a 20 minute walk but Liphook station is very small and there were no cabs. Uber wasn't such a common thing back when I went and you might be able to arrange that now. But if you're going to walk all day, what's an extra 20 minutes?