My husband is English and we go "home" every year (can't wait -- I love the cooler weather over there!). So I've been there with our daughter every year since she was born 12 years ago and have certainly been with her there when she was your son's age!
It would help a lot if you could update and say where you will be most of the time. Being in London or any large city is very different from being out in a town or village. Our relatives are in Devon and Dorset and we always found that there are lots of indoor play places for kids (and I do not mean McDonald's!). Many a farm has converted to some combination of a barn that's now got a ball pool, plus a petting zoo or crazy golf....There are also many, many gardens and parks, everywhere including the big cities, where you can take your son to run play.
Go online for the area where you'll be staying and look up "children's play areas" and see what you find.
Also, many historic sites cater very nicely to kids and have dress-up rooms and hands-on items they can touch. Almost every historic site seems to have a "kids' trail" as it's called -- a sheet you can take around and have your son find things: For instance, a typical kids' trail for a child his age at, say, a historic house, might tell him to locate the suit of armor in the great hall, or the wind-up toy mouse who lives in the kitchen, etc. (They'll make it pretty easy!) Having a child focus on finding specific items in each room allows you as the adults to see every room too--everyone wins; he's occupied and you get to go to the historic site if that's your thing! Often there is a reward if the child (and you, writing on his sheet) return the completed trail guide when you are done. British sites do this brilliantly! Ask for the trail that is for his age group or if they only have ones that seem a bit old for him, take one anyway and just adjust it yourself as needed.
Many historic sites also offer special events and hands-on activities for children all summer long. If you want to go to a specific site, check online for the date you would go and see what's on. I recall our daughter at about age five having a great time decorating a large posterboard "shield" with her own made-up "coat of arms" at one castle. Another time she made a wreath of flowers to wear in her hair. Yet another time she and a bunch of kids helped a queen get dressed! I can't say enough good things about how well these sites do all this.
Be aware that when it comes to food, most historic sites, museums, large gardens etc. have cafes --not mere cold-sandwich places but real cafes serving usually at least a few hot dishes. It's great for lunches when out. But the flip side is this: If your young child and you need dinner at, say, 5:30 or 6:00, you may find -- if you are anywhere outside London or a large city -- that you are out of luck; many restaurants and pubs do not even start dinner service until 7:00 or later, or smaller restaurants in smaller towns don't serve dinners at all but only breakfast and lunch. Be aware of that and keep some food with you. We did that a lot when our daughter was young, knowing that we might not get restaurant food until after 7:00.
Take boats when you can. If in London, try to get tickets to the London Eye Ferris wheel if your son is OK with great heights (but get the tickets before you leave the U.S. -- it books solid!). Look for age-appropriate kids' plays too, especially in London if you are spending time there. Try to get a copy of the magazine Time Out London in the airport as soon as you step off the plane; it's a weekly guide to what's on and includes children's events and entertainments.
I hope you have a wonderful stay and that you can take him back a lot. Our daughter is never bored even though we stay in a small and rather slow town where the in-laws live--she even thinks it's fun to go to the grocery store there because it's all different. She's very proud to be half English and to know "her" English home area really well now. Have a blast!
