Congrats on what can be an amazing experience if you’ve never been to Europe. You are going when the exchange rate is very good. You may want to confirm that May is still the shoulder season. If not, April lodging rates will be much lower. Once you decide what you want to see / do, you can then determine if there is a tour that covers those things.
My wife and I went a dozen years ago, but we won’t be tour people until at least our 70’s, if then. For us a significant percentage of the fun was deciding what to see and where to go. We both spent a few nights researching (cost, location, time required) and ranking the sites that we each wanted to see. We then spent a little time each night discussing and narrowing down the sites. It was a packed two weeks that we will never forget. We actually hit everything that we planned to do, but weren’t allowed into two of the castles due to the queen being in residence, or about to be in residence. We used Priceline and Betterbidding to pick a great rate on a great hotel next to the tube in London, took a train to Bath for a half-day, rented a car and drove to the Cotswolds, Nottingham, York, Inverness, Edinburgh, and lots of smaller places along the way. We met tons of wonderful people, saw historical sites with a whole new perspective, and thoroughly enjoyed the entire trip, including our 1.5 “rain days”. The people of England, including London, were much warmer and friendlier than we expected.
I can be kind of thrifty :-), but we did not pinch pennies on this trip (nor did we spend like a drunken sailor). This was going to be, in all likelihood, our only trip to England, and I did not want to constantly be calculating the exchange into dollars. I don’t recall exact numbers, but our two weeks were way under $3000, including airfare. We stayed at one B&B for 2 nights, one 3 star hotel for 3 nights, one 4 star for 4 nights, and a five star for 1 night. The remainder were with family. To get a very rough idea of costs, just figure out how much the trip would cost you in comparable areas in the U.S. and then multiply it by the exchange rate.
Finally, although we saw tons of sights, the two that stand tall above the rest are the British Library and the Cambridge American Cemetery.
Whether you do a tour, or do everything on your own, have fun.