Giggleswick

Just an idea, but...I see from another post of yours that your kids are in their 20's, old enough to make it into London from LHR on their own. So, assuming you'll be staying in London, and unless you are set on sharing a taxi or something, have you considered heading into the city and having them meet you at your accommodations later in the day?

If you're lucky, your room will be ready when you arrive. If not, you almost certainly can leave your luggage there for free (you could email in advance to check that). When landing at LHR after a transatlantic flight, I want to get out of there ASAP. Unless you have access to a very nice airport lounge, almost anything you can do in the city--from having a decent meal, to sightseeing, to just sitting in a hotel lobby--is likely to be more pleasant than 6 hours in the terminal.