OK, you have a good start on a great trip. As suggested, book your flights as a “multi-city” flight (priced almost like a roundtrip), with your outbound flight US to Dublin and your return flight London (LHR for Heathrow) to USA. Don’t include the transport from Dublin to London; leave that for later when you understand the options better.
Remember that if you depart the US on August 25you will arrive in Dublin on August 26. (I mention this because even after 20 trips to Europe I sometimes forget this fact). So spend 3 nights in Ireland, and travel to London or elsewhere in the UK on August 29.
This is great timing, as it will put the last part of your trip in September, which is our favorite time in London. That is largely due to the “Totally Thames” month-long festival. This is a mix of cultural, historical, musical, athletic, and other celebrations of the role of the River Thames in London’s history. It is a festival for locals. Not that it doesn’t attract some tourist (like us) but it seems no more than the usual number of visitors at that time. And prices do not seem to be elevated above normal.
https://thamesfestivaltrust.org/ Use the “what’s on” button to explore, but remember the schedule for September 2022 is nowhere near complete at this time.
You will have roughly a week left; I suggest you explore first (maybe Bath, or Salisbury, a village near Highclere Castle, or ???) and save London for last since you will need to spend a night there before flying out on Sept. 6.
One option you might consider is to take the ferry from Dublin to Holyhead (in Wales) instead of flying from Dublin to London. The ferry takes roughly 3 hours (more or less) and there are are at least 5 sailings per day. From your landing at Holyhead, you can take a direct train to London’s Euston station and be there in 3 1/2 hours. But if I were doing this, I would go only as far as Conwy in Wales—-less than an hour’s journey by train from Holyhead—-and overnight there. Rick Steves describes Conwy in his “Explore Europe Wales information page as “one of Britain’s most pleasant small towns” and “one of the most purely delightful British towns of its size; and Conwy Castle as “Wales’ best”. That’s pretty convincing. (Full disclosure—-we have not been there yet).
The next day continue on the train directly to London, and then change trains for your chosen destination outside London, whether Bath, Salisbury, or other). There may be other routes to those cities from Conwy than going all the way into London, but all the ones I saw seems to require lots of train changes. But you’ll have to change train stations in London (likely to Paddington or Waterloo for those particular towns) and perhaps changing trains outside of london is preferable. You can figure that out later too.
Whatever you decide, leave at least 3 or 4 nights at the end to spend in London.
Trip insurance: rather than getting recommendations for a specific company, I suggest you go to one of the websites that helps you compare several different policies to get the one best for your needs. I use either insuremytrip.com or Squaremouth.com and don’t really have a preference for one or the other. Most policies provide both trip cancellation and medical/evacuation coverage, and you should have both. The cost of the insurance is based on a combination of trip cost and length, your age, and maybe other factors such as where you are traveling.
For trip cost, you calculate that based on the total amount of NON-refundable costs. Generally this includes airfare and any tours you have booked, as well as non-refundable lodging if you book any of that (not advised these days). You won’t know this until you actually purchase your flights so you have time to read up and learn more about this. Generally you must purchase within 2 weeks or so of your first non-refundable payment to have coverage of pre-existing conditions.