Three weeks in England and Ireland itinerary advice

presumably Day 4 means "Cotswolds OR Stonehenge ?" They are in opposite directions from Bath and you won't be seeing both in a sensible day.

Day 6. Have you checked flight schedules? I've done a quick scan allowing for travel from Bath and can't see anything from Heathrow or Bristol that doesn't require a VERY early start from Bath if you want a clear afternoon in Belfast. I've not bothered checking other than Heathrow for London departures because the journey from Bath to the other so called London airports is so long. Cardiff has a noon flight direct to Belfast which would be easy from Bath but you won't be in the city until after 14:00

While London is certainly worth a lifetime of multiple visits, you’ve seen the major sights and don’t have much you say you want to do there — just a sports stadium and eating, which you can do pretty much anywhere— while your Bath days are so overpacked, you might want to consider just going straight to Bath on arrival and skipping London, or at least giving Bath one of your London days. Not sure what you mean by “Titanic Belfast stadium.” Do you mean the Titanic Belfast attraction (which is well worth visiting), and/or a stadium somewhere in Belfast?

In Cork, I would add a day trip to Cobh as a follow-up to the Titanic in Belfast. The train ride is very scenic. The bus to Blarney is easy and quick. Sin é for traditional music. https://corkheritagepubs.com/sin-e/
Check on which Dublin attractions require advance reservations (probably Book of Kells, Guinness and Kilmainham).
The HOHO bus in Dublin isn't very useful, as the traffic is crazy and things are generally quite close.
Paddywagon offers day tours from Belfast, Galway and Killarney. https://www.paddywagontours.com
You might want something smaller for the Ring of Kerry. I can't help with that, but search the forum, as it comes up quite often.

Mad Max has a day tour from Bath that goes to both Cotswolds and Stonehenge.

I hope you report back regarding which sports venue you choose to tour, that's right up my alley as well. I did it often when I was visiting US cities and I've seriously considered it several times in London but haven't pulled the trigger yet.

I wasn't a fan of Borough Market. We were there on a cool, windy day on a weekday of the 3rd week of November last year and it was still shoulder to shoulder people. My wife is only 5' tall and was miserable.

Consider the sunrise tour of Stonehenge, it allows you to walk within the stones which I thought enhanced the experience. On the way back you took take the bus that stops at Old Sarum which is where the original Salisbury was located. I'd also seriously consider spending the day in Salisbury and taking the Tower tour of the cathedral.

Some thoughts from me:

-If thinking of visiting a sports-related place, I would also consider the Oval- there are still matches you can attend in September and it's such great atmosphere. If you're thinking of seeing a football match I would also recommend considering Craven Cottage (Fulham)- they are not quite as famous as some of the other London teams but they have such a unique stadium right on the river, and it's also a really good atmosphere. And it should be easier to get tickets than some of the bigger names.

-I agree Cotswolds and Stonehenge are too much for one day- I would select just one. You can combine Stonehenge with Salisbury which has a very beautiful cathedral.

-Most of the Causeway Coast tours include everything you mention, altho I've heard they no longer stop at Carrick-a-Rede so you may need to check that.

-When you say Titanic Belfast Stadium, do you mean the SSE arena? If you want to see an iconic Belfast sports location I would consider Windsor Park instead.

Good luck with your planning!