Staying near the airport that first night makes sense- since there may be no public transportation options on Christmas night to get into the city. You can wake up refreshed on Boxing Day and get into the city then.
You could take a taxi to your apartment or hotel. Crunch the numbers…which cost less…the taxi or the hotel?
My DD was 15 when we went…and we went on a tour of the Royal Albert Hall. It was terrific.
The good news is that you have lots of time to figure out your hotel & transportation for Christmas.
Transport to Heathrow at the end of your journey can be done via Heathrow Express out of Paddington Station (10 - 25 pounds depending upon time), bus, taxi or the Piccadilly tube line which we have used. It is only about 6 pounds and takes 40 minutes from central London. Just make sure you note your terminal number. You can also cash out your tube Oyster cards when you arrive at the terminal.
Try the trip advisor London forum. https://www.tripadvisor.com/ShowForum-g186338-i17-London_England.html
Hi Big Chef. Yes we chose that way. We are landing to Brussels at 8:00 am Christmas morning. Spending the day with a local friend until 4:00 pm and then take the flight to London and land in London at around 5:00 pm. By the way we chose it that way so that we could spend some time with our friend in Brussels.
@kajon the tube won’t be running Christmas Day. Will the other train and bus public transports be in operation?
I have been to London a half a dozen times and I never did the tour of royal albert hall (which sounds like a terrific thing to do). Which only proves that, like other big cities (NYC, etc.), there are an unlimited number of great things to do in London!
Oh, I’m getting so excited for my trip to London with my parents this summer! 
I have an online friend and a nephew who are both researchers at Oxford, so we will visit them, too. And I get to hear my dad speak at an engineering conference.
@MaineLonghorn - ^^^ My youngest son went to grad school at Oxford. What a fun place to visit.
@thumper1
@ArdenNJ j - oops on post #22
I wrote at the “end of your journey” and meant that as - when you are returning home and all transport is up and running.
A good West End show would be The Play That Goes Wrong (if it’s still playing). Hampton Court has a special Elizabethan Christmas event that time of year. It’s easy to find a day trip out to Stonehenge.
We rented a two bedroom flat on a quiet street just a block or two from the Sloan Square tube station. While pricey, being able to make our own breakfast (and many nights dinner) meant we stayed within our overall budget, and we enjoyed shopping in nearby markets and felt less like tourists. The agency we used, Farnum & Christ, is based in the US and was very easy to work with. I was a bit leery of using something off the internet but reassured when in the initial stages of planning learned that one of the firm’s principals was a friend of one of my college friends. https://www.farnum-christ.com/index.php/search-by-neighborhood/sloane-square-chelsea/2br-sloane-gardens-york.html
I would check to see if the Heathrow Express train runs on Christmas Day as that is the easiest option to get into London. Last time I was there and it is more than 10 years ago initially stayed at a Hilton right by Paddington Station and then moved to one of the Thistle Hotels, not sure if still in existence but it was a great location opposite Hyde Park on the northern side, very centrally located and still walking distance to Paddington Station for the return to Heathrow.
I would definitely recommend the Tower of London as they will be pumped that they had visited when they study Global History, same with the British Museum and all their collections. There is also a very interesting Museum of London right near St Paul’s Cathedral that gives an excellent history. Both St Paul’s and Westminster Abbey are definitely worth visiting. I loved day trips to Greenwich to see the prime meridian and the other sites and to Hampton Court. London Eye fun to do at least once. The Tate Modern is a great space and fun to visit. So much to see and do in London!
I know London really well - being British and all - and all of the suggestions have been fantastic. ALL museums are free!! Certain exhibitions in the museums may cost but general admission is free. Check opening times around the Christmas period. Lots of things may be closed on Boxing Day - the 26th.
However, all the lights will still be up and all shops will be opened as the sales will have started. The Christmas tree in Trafalgar Square is worth a wonder past and The National Gallery (Art) is a good shout. Loads of recognisable paintings. A hidden gem if you like paintings is the National Portrait Galley tucked behind the National Gallery. Doesn’t take too long to get around and has some very recognisable people in it. (Lots of British history so some may be lost on you - but it’s still free and warm!)
It will be cold. Not sure where you are starting from but British winters while they might not look cold 0-5 degrees centigrade, it is a damp cold and it could well be raining and blowing a gale. But it could also be very pleasant. There’s no telling.
Regarding accommodation I would definitely recommend AirBnB. Most probably the same price as a hotel but eating out will be pricey. This way you have the choice of eating in and keeping the cost down. Food to try is definitely Fish and Chips. Do take away and sit in a park and eat them out of the paper. Splash out on afternoon tea.
There’s the fantastic B&B teas - will try and find the link - based out of Convent Garden that does one on a red double-decker bus going around the sites. But if you want to experience the full blown affair look up Harrods/The Dorcester/The Ritz/The Waldorf. (Maybe good for 26th as they will be open.) Something really different. You can order coffee! At the other end of the scale go to a ‘greasy spoon’ and have a full English breakfast. We picked one up for £6 last time we were there. Eggs, bacon, sausage, tomatoes, beans, mushroom - maybe black pudding, fried bread and toast and buckets of tea! If you are a little out of the centre of London these will be easier to find. Basically workmen’s caffs.
However, coming to the best thing ever to do in London. It does now cost but so worth it and much better than the London Eye - which also costs - is a trip up St Paul’s Cathedral. Climb I don’t know how many steps to the whispering gallery a piece of fantastic engineering. Have your sons stand opposite each other and whisper into the wall. Then climb even more steps right to the top and you get the best view of London and you feel that you have earned it. Then of course you climb all the way back down. The catacombs - very clean with a cafe - is also nice. This is my recommendation to all visitors and so many say it was the best thing they did!
As a ps. We are members of Home Exchange and if you want to go that route I bet you could find some amazing accommodation over Christmas. Many people are travelling to family across the country or even looking for a trip to the states and for an annual fee of 130Euros it’s a steal.
There are no trains (including Heathrow Express) on Christmas Day or Boxing Day. Tube starts on Boxing Day.
See https://tfl.gov.uk/status-updates/major-works-and-events/christmas-day-2018 and http://www.nationalrail.co.uk/service_disruptions/172154.aspx for details of each day during the Christmas period for last Christmas. There are also some more limited closures around New Year.
I’d agree with the idea of staying at the airport on Dec 25th and going into London the following day. I’d expect most tourist sites will be closed on Dec 26th, though you can still walk round many interesting places.
Two thoughts in no particular order:
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What are the logistics of customs/immigration for having a flight booked through to London, leaving the airport in Brussels, then turning around and re-entering the airport in Brussels with no luggage several hours later? Curious for a trip I may be planning this summer
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You know your family best and what your 13 yr olds will tolerate for where to stay on December 25th. They may be tired from long flight and stop in Brussels - and just want to crash when you get to Heathrow airport. On the other hand, they may just want to keep going to your final accommodation and only unpack once.
If you decide to go with a hotel, there are several Marriotts within walking distance of the Bond Street tube stop. That is also a great area for quick meals that will satisfy a teenage palate 
I think mine were a tad younger when we took them. We rented a flat near Marble Arch, great location. The girls loved the ethnic diversity in London, especially the Arab shopkeepers nearby who kept giving them free candy. This is also near/walking distance to one of the Heathrow busses, near a tube station and cinema. I’d stay a bit further IF that area is charming, not just for less expensive.
They also got a kick out of Mme Tussauds and the British Museum. The Beefeaters at the Tower were sweet to them, but mine are girls.
Our flat was via HomeAway, one bedroom but a pullout couch. Gracious, classic building. Taxis nearby, if needed. We cooked on our own, some eves. Renting an apt, even small, is a wonderful way to feel like a local. To me, that’s priceless.
We were there in winter, but I think Feb. Anyone used to northeast winters shouldn’t find it too cold.
Thanks again for all the suggestions. We booked this hotel/apartment right next to St Paul’s Cathedral.
Do you guys approve? $251 per night 2 Bedroom apartment. Nothing luxurios but has great reviews and seems like instead of a hotel we might enjoy an apartment better.
Look wonderful! Love that area!
You’ll be close to the museum of London, which was one of our favorites.
Perfect for your trip up the cathedral!! And a great walk across the Thames to Tate Modern - always a good show - and free.
Not sure how long you are staying but organise an oyster card. This is the transport card. Children under 16 pay £15 deposit and then all travel is free!!
https://tfl.gov.uk/fares/free-and-discounted-travel/11-15-zip-oyster-photocard
Poke around the site and see what is best.