London for 12 days

You are traveling during the height of the tourist season and only a few weeks from now.

Expect long queues and hordes at the Tower of London, Westminster Abbey, Changing of the Guards, Parliament Square, Trafalgar Square and Churchill War Rooms. I’d try to make advance reservations for the Tower and War Rooms. I’d ignore the Changing of the Guards. I’d go early to Westminster Abbey and take the Vergers tour.

Camden Lock and Brick Lane are fun markets.

Check online for same day 1/2 theatre tickets and buy them at the Leicester Square ticket booth. https://officiallondontheatre.com/tkts/

Review www.walks.com and see if any interest the family.

Tower of London-go when it first opens.

You may also want to look into London Walks, they are a cheap and the guides are extremely knowledgeable

A fun "out of town" day trip is to Dover. Take the train and visit the Castle as well as the Dover War tunnels. It is quite interesting.

If the boys like Harry Potter, they may prefer Oxford :)
Definitely book your theatre shows ahead of time, to avoid disappointment & get seats together.
The boys might really like Camden Market, it's not boring :) When I lived and worked in Hampstead, it seemed like Camden Market was open every day, but it's probably best to check on their web site first.
If they are really considering studying in the UK, use this:
https://www.studyacrossthepond.com/
GREAT free service for students in the USA who want to go to the UK. I've been using them for a year and can go to grad school (now that my daughter's married) this Sept cuz of all their help.
When I went to Buckingham Palace, there was also a ticket with timed entry to the Queen's Gallery. It had some great art, but it was small and not that interesting to me (someone who loved artists from Amsterdam would like it), and the timed entry thing added stress to our day. I would suggest just regular tickets to Buckingham Palace's state rooms. The line for that moved fairly quickly when I was there, but that was a while ago.
I overheard a guard at Windsor Castle saying since someone had not bought their ticket directly from their box office (online or in person), but bought it from another service (maybe a london pass kind of thing) that they were not allowed to enter later in the day, they had to be there before 3PM I think is the time he said. So in case you think of going there, earlier in the day is better (plus there's a changing of the guard there sometimes). I don't know if that rule holds for other sights too or not. It might.
British Library Treasure Room is great for a day where the weather is bad or when you've tired yourselves out, and might need one of the many comfy benches. It's right near King's Cross, with Platform 9 3/4 and their new gift shop. St Pacras hotel is next to that, and then British Library. The pub across the street from the British Library was really good when I went.
So much of travel is identifying what YOU all really specifically love to do and to see. Enjoy!

Thanks for the responses.

Any insight on the day trip ideas?

Check out http://www.londonwaterbus.com/ for a nice look on the back side of London from the water level.

http://www.jasons.co.uk/ is another option, but the waterbus can drop you off inside the zoo.

The city of Winchester might suit you for a day trip. Train from Waterloo takes about an hour. Winchester Cathedral dates back to the 11th century, and there is a small museum close to the Cathedral about its history. There is a weekly market, I think on Fridays, but you can verify that with the Tourist Office. It is an easy walk from the train station to the town centre.

Other day trip ideas:

• Windsor
• Greenwich (yes it's London but so much to do and see there so it's like a day trip)
• Hampton Court Palace

In 5 trips to London I have yet to see Cambridge, Bath, or Stonehenge so can't advise on that. Oxford was bustling with students (obviously!). I loved my day trip to Winchester, King Alfred's capital - the cathedral, Itchen River, the historic mill (I went on a demonstration day so they were milling), High Street, Wolvesley Castle (ruin), Winchester College. Easy train from London.

Good luck!

Oxford makes a great day trip from London. The Ashmolean Museum offers an eclectic collection. Visiting the many college grounds and gardens will make for a full day. Blackwells Bookstore is amazing. In addition to the more well known colleges and gardens, the view from Lincoln College down Turl Street is very photogenic.

Another worth while day trip from London is Bletchley Park, which has a fascinating museum covering the history of the work to crack the Enigma machine code during WW2, along with one of the sets from “The Imitation Game” still on display.

I think you would enjoy Dover castle as a day trip. If boys are Harry Potter fans, you may not be able to get tickets to studios in Leavesden now but should try. Cambridge has the Eagle Pub where Watson and Crick announced that they had determined the structure of DNA plus Kings College chapel. I assume you're going to Bath to see the Roman Baths. If not, it would be easier to get to Stonehenge from Salisbury because there is a special Stonehenge bus. There is also a copy of the Magna Carta at Salisbury Cathedral. Allternatively,, if Mad Max has an afternoon tour to Stonehenge from Bath. York would also be a great day trip.

Your trip is coming up very soon, so you may have trouble snagging discounted train tickets for some of the destinations being discussed. I suggest going to nationalrail.co.uk to check fares. That may help you prune the list a bit.

I'm going to sound like a stuck record but......come to Portsmouth! An hour on the train from Waterloo (or 1/2 hour longer but cheaper from Victoria) straight to Portsmouth Harbour where you'll find the HIstoric Dockyard minutes walk from the station https://www.historicdockyard.co.uk. If you check Groupon (or other voucher sites) there are pretty much always discounts on tickets and annual passes. There's plenty that will interest your boys, and yourselves. Nearby Gunwharf Quays https://gunwharf-quays.com offers plenty of choice for somewhere to eat for picky teenagers (Nandos is THE choice for pretty much every omnivorous teenager in the UK for some reason).

If you want to explore further I can give advice on some places all within walking distance that will give you a feel of Portsmouth's history.

An easy hour's ride on a regional train from London will take the teens to Bletchley Park. It is the isolated estate where the WW2 codebreakers helped win the war. Mathematicians, crossword puzzle aces, and very loyal secretaries helped sort out the Nazi radio messages, helped by a huge clattering machine that is sometimes seen as a forerunner of the modern computer. The Bletchley museum is fairly new since the vows of secrecy kept historians at arm's-length for many decades.
https://www.bletchleypark.org.uk/

I do recomened if your boys are interested going to the Museum of London. Its more about the history of london. Keep in mind its rather small but because its not one of the bigger museums you don't have to worry about long lines. Its a bit hands on but still enjoyable (i am in my 20s and i love it).

I second what others are saying about the tower of london and war rooms and booking in advance or getting there early! Made the mistake of getting to the tower when it busier and was in line for awhile.

I did a bus tour that had stongehenge and bath with Evans and evans tour. They do coach tours to them. Bath is about 2hrs away from London and Stonehenge is about an hr and 42 minutes away (current calcutions so i might be off). Somone mentioned winsdor and evans did have that as part of the tour. All three were great! The admission to those locations were included and our tour guide was super nice! But i'd recomened checking out their website see what tour is perfect for you guys here: evans andevans day tours from london

One fine note: Its overstated but the british museum is awesome! I kept going back and finding things i had not seen before. Its still one of my favorite museums and when i am going back to London i'm going to be there! They have late friday days where they are open to 8, have tours of different rooms and tables where you can touch objects (wish they announced what room and what time.). However certain rooms are going to be busy (anceint egyptain rooms and where the rosseta stone is). Keep in mind though they have security checks of bags. Also don't use the main great russel street entrance but rather the Montague Place because you will get in quicker!