Question on Afternoon Day Trip from London before Next Day Heathrow Departure

Richmond would be very simple. You won’t even be leaving London. You won’t be reliant on closing times either - it’s a nice place to wander with plenty of places to eat and drink.

For me all those options will be too rushed apart from Richmond. If that's something you're interested in then just go to Richmond and stay there and see the "sights". It will be a nice location for your last night with no time pressures and an easy transfer to LHR the day after.

I think Cambridge and Oxford would be a bit of a time stretch.

Richmond is doable, and you could have dinner at The Prince’s Head, which is the pub featured on the show (and their food is delicious—I ate there last year). The town is quite lovely and very walkable. There is also Ham House and Richmond Park, which the kids would love, I'll bet. And you can walk the Thames Path that gives you great views of the river at that point. Here are the directions I have from my trip last year:

Walk Thames Path. The trail starts from Richmond Station and leads you through Richmond Green, past the Old Court House, and down to Asgill House along the Thames River. Continue on the riverside path, passing underneath Richmond Bridge and along the Rotary Gardens. Just past the bridge, there is a River Thames Visitor Centre. Follow the path through Buccleuch Gardens and Petersham Meadows, continuing around the riverbend past Ham Lake and Ham Lands. Ham House and Garden is on the left. Along one of the loveliest parts of the Thames, is tiny Corporation Island (by the White House Pub), colonized by flocks of feral parakeets*

The other possibility would be Hampton Court Palace. If you get a 1:30 pm train out there, you should be there by around 2 pm, then it's a quick 5 minute walk to the palace, and you can easily see most of the inside in a few hours. Then head out to the Magic Gardens, which should be really lovely in June. You can also have a meal at the Tiltyard Café, which is in the middle of the gardens, and send the kids off to try their luck with the maze.

Richmond is an excellent choice.

Keep in mind that sometimes the Eurostar does not operate on time. While departing St. Pancras in March, the Brussels bound train was already delayed almost an hour when my train to Paris left on time.

Another option is to spend your last night in Windsor.
After arriving in London (Sunday July 5 around 12pm) you would take the train immediately to Windsor. Drop your bags at a hotel, then see Windsor Castle and walk around town.
Stay that night in Windsor.

Windsor is a lovely town with lots of good restaurants. The Thames River is right there, with boat rides you can take.
Take a taxi the very short distance to Heathrow the next day
to the door for your correct terminal.

Hampton Court Palace (we have a membership so this one is free for us) - a bit worried it's too late in the afternoon - last admission time appears to be 4:30 (though Magic Gardens open until 6).

Another option is to take the train to Hampton Court Palace as soon as you arrive in London. They have luggage/bag storage at Hampton Court, so you could stash your stuff, tour the Palace, have a wonderful day as described by Mardee, then take a taxi to Windosr.

You then would not have time to do Windsor Castle, quite obviously, but would have time to enjoy the town, taking walks around the medieval streets, enjoying a good meal (lots of restaurants) and stay the night at a hotel there.
The reason being that Windsor is a nicer place to stay for your last night than an airport hotel.
I have stayed at The Castle Hotel and The Macdonald Windsor hotel there and liked both.

Good luck deciding what to do. You have lots of choices.

still looking for reasonably priced hotel for 3 adults/3 kids (2 rooms) hopefully near St. Pancras

Of all your options, this would not be my pick. If you end up deciding to stay in London to go to the theatre that night, stay somewhere near your theatre. Or walking distance from there....convenient for when you exit after the performance.

My opinion of Richmond.....lovely and an excellent choice.

Mardee's suggestion of Richmond is a good one.

(I agree that you just don't have time to do Cambridge or Oxford.)

If staying in Richmond, walking in Richmond Park and along the Thames Path would be lovely.

There's a boat leaving from the riverfront that will take you to Hampton Court. See Hampton Court, then take the boat back to Richmond to stay for the night.

I suggest the Richmond Hill Hotel. The Petersham is nice, but a bit pricey.

Lots of good restaurants and cafes in Richmond.

You could take transportation from Richmond directly to Heathrow (taxi or bus) on your day of departure.

You could book something Richmond that is cancelable. Then if after your initial time in London you realized there was something more in London you wanted to do you could change to a central London hotel and do that when you come back.

Thanks all for the very helpful and thoughtful tips - sounds like my wife and oldest sons may well get their Ted Lasso fill :) I also read recently where they appear to be possibly starting filming either in May or June for Season 4.

In any event, i'm going to look in Richmond area (with Hampton Court still in play), but also Windsor as well, just to see what options (hotel availability and pricing is). With 4 kids i'm trying to limit us to 2 rooms, which is alot to ask in some places around London :)

Much appreciated for the helpful feedback.