That's a lot of things to pack into 3 days. I would really prioritize and aim for one maaaayybbe two "big" sites a day, and make sure to build in breaks to keep everyone's spirits up. British Museum is not air conditioned and could be really hot and stuffy that time of year, so I would go there first thing in the morning.
If you're going to Globe Theater, Borough Market is nearby and is a good place to get lunch or a snack.
My son loved doing a boat tour on the Thames. It's a good break from all that walking while getting to see the city from a different perspective. And they sell beer on board (for not crazy expensive), so parents get a break too :)
If you're choosing between St. Paul's and Westminster, I personally prefer Westminster because of the history, but your kids might prefer St. Paul's for the dome climb. Tickets ahead of time for Westminster are crucial, St. Paul's I think you'd have no problem buying on the spot.
Also if your kids are into playgrounds, there's a good one near the London Eye called Jubilee Gardens Playground. It has lots of rope structures for climbing and might be good for older kids. Our kiddo also really liked splashing around in the Princess Diana Memorial Fountain in Hyde Park and the Diana Memorial Playground, in adjacent Kensington Gardens, can get crowded but also has stuff for older kids.
Have a great trip!