3 nights in London with 4 Teens, Ideas?

Well, that's a steep order. But here's a couple of ideas. Good luck!

Harry Potter - Platform 9 and 3/4
https://www.thecrazytourist.com/25-best-things-to-do-in-london/21/

See the Speakers at Hyde Park
https://www.thecrazytourist.com/25-best-things-to-do-in-london/

Broadway Market
https://www.airbnb.com/things-to-do/london

Imperial War Museum
https://www.lonelyplanet.com/england/london/attractions/imperial-war-museum/a/poi-sig/370861/358914

First of all, the Harry Potter train -- if you are referring to the steam train from Fort William to Mallaig -- is indeed a long trek. If you only have three nights, there's no way. You'd need to plan a separate trip to Scotland to do justice to that item.

Funky offbeat unique things could include Sir John Soane's Museum, which is more history than art. I stumbled upon it when I was headed for Dickens' home in Lincoln's Inn Fields, and found it fascinating.

London Walks has some offbeat walking tours (Jack the Ripper, Ghost Walks, Harry Potter, Beatles & More...). You can even go on a morning jog/run with a London Walks guide. The nighttime walks that stop off at pubs may have a minimum age, so be sure to ask ahead of time. http://www.walks.com/

With my teens we really enjoyed the Tower of London, including the ceremony of the keys, Hampton Court Palace, Spitalfields Market Platform 9 3/4, London Eye and London Walks. One of the walks we particularly enjoyed was the Harry Potter walk which was interesting for both adults and teens. Also visited Sherlock Holmes museum which was ok but my son really wanted to see it.We were in London in March a few years ago and got a great rate at the Thistle Euston which I thought was a good location for getting around.

If they like history, I second the Imperial War Museum; they'll love it, and could spend hours there. There's also the Churchill War Rooms: http://www.iwm.org.uk/visits/churchill-war-rooms and Museum of London, for the city's history: https://www.museumoflondon.org.uk/museum-london

Some other non-art museums are the London Transport Museum: https://www.ltmuseum.co.uk/ and the Science Museum: http://www.sciencemuseum.org.uk/

As for foods, would they want to try fish and chips, or is that not "exotic" enough for them?

What about Camden Street Market. My two teenage granddaughters enjoyed walking through it.

For affordable lodging I recommend The Jesmond Dene which has some larger rooms which may work for your family. It is clean and includes breakfast when you book on the hotel website.

Horniman Museum, Brick Lane, Spittafield's Market, Camden Lock Market, looking for the door to Harry Potter's Leaky Cauldron at Leadenhall Market, Imperial War Museum, Jack the Ripper tour with London Walks, the Treasure Room at the British Library, The Shard, food court at Harrods are some options the four teens might find of interest.

Take them for lunch at the Art Deco Blackfriar pub. I'd also get tickets at the 1/2 price ticket booth for an agreed upon play or musical.

Even thought they don't like shopping you should still visit Liberty. Doubt they've seen a store like it.

Lastly, if they like history then a Westminster Abbey tour is a must.

It isn't clear what time of the year your trip is so I can't say what will be seasonable - like Rhubarb and Vale of Evesham asparagus currently, strawberries to follow - but if it is the right time in the week the teens may get a kick out of the most incredible food market in the country. Have a look at Borough Market - http://boroughmarket.org.uk/

Any of the teens or you a musician or fan of classic instruments?

There aren't really any foods that are unique to London, jellied eels would probably be the closest to a unique food although these can be found elsewhere. Another traditional London meal is pie, mash and liquor (the liquor being a parsley sauce). A full English breakfast can be found everywhere but varying wildly in quality. To make it different to an American breakfast of bacon eggs be sure to have black pudding (a blood sausage) and the bacon is likely to be back bacon (Canadian bacon). I find the breakfasts at Bill's restaurants to be of good quality and not bad for a chain and there are quite a few scattered across London. Of course there will be a lot of very good independent places but unfortunately I don't know any to recommend. If they liked the pickled herring they may well like cockles, small clam like shellfish served in little pots of malt vinegar.

Most of the markets that aren't boutique or farmers markets are usually full of cheap tat and really aren't of any interest to anyone who isn't looking for fake goods, vaping equipment or cheap polyester clothing, it'd be the outdoor equivalent of a dollar store.

Pubs are a real gamble, there are good ones, bad ones and terrible ones. It depends on what you're looking for in terms of a pub experience. Pubs are in decline in the UK due to a number of reasons, a change in drinking culture, the smoking ban and the availability of cheap alcohol in supermarkets. Expect to pay at least £5 for a pint in a London pub or less if you go to a chain such as Wetherspoons.

Have you checked yha.org.uk as they have family rooms and no age restrictions. They also have lower rates for under 18s. Oxford Street, London Central, St Pauls and St Pancras are all in the heart of the action and Earls Court is a short tube ride away.
You are right, there is so much to do in London that you will only get a taste. It can be great just wondering around and enjoying the everything.
There are also plenty of day trips within an hour by train. Brighton has the Royal Pavillion which is both historic and funky. Brighton as a whole has a funky feel combined with the seaside feel and a very dated amusement pier. Windsor is more traditional with the castle, great park, and Eton college.
Good luck with accommodation!

Brixton Market on a Saturday is fun. More down to earth and lot's of eateries.
Highgate Cemetery is very historical and very cool. The West side ( must sign up for tour) is by far better than the East.
About food - find some cornish pasties and picnic in one of the many beautiful parks.
A boat ride to Greenwich on the Thames is a good way to sightsee.

When is your trip?

I was going to suggest mudlarking ( searching for artifacts at low tide on the Thames) but the activity has been recently (and appropriately) curtailed and requires a permit. However, Walks of London offers a walk along the banks that features archeology and history.

http://www.walks.com/our-walks/thames-beachcombing

You can go down to the foreshore on your own afterward, just don't dig.

Another thing they enjoy is a play at Shakespear's Globe, standing below the stage---I forget what those tickets are called but people seem to enjoy it.

groundlings

To avoid confusion: LOL

groundlings = the name of tickets where you stand through the the performance in front of the stage at The Globe.

The Groundlings = an improvisational comedy performance group that began in Hollywood, CA in 1974.
Still going strong.

Pubs often serve full menus now, occasionally good enough to call themselves gastropubs. Sunday "roast" is easily available, with a joint of beef or lamb (or, as an alternative, roast chicken) with all the fixings including Yorkshire pudding. It is eaten as the midday meal and reservations may be necessary. An old favorite any time is bangers and mash, with the sausages atop mashed potatoes and plenty of industrial-strength gravy. There was an old joke about English sausage not being allowed in the European Union because it had so much filler and so little pork as to not technically qualify as a sausage. Ignore such spoil-sports, but plan on a nap after the meal, letting the kids run it off. There are plenty of on-line pub guides, including the on-line events magazine Time Out. But an early inspiration is http://www.camra.org.uk/ which started a half-century ago to promote "real" English ales at a time little breweries and pubs faced extermination.

You may have issues due to drinking age in pubs. Some traditional "drinking den" pubs will not allow under-eighteens in at all, or may not in the evening. It is illegal for under-eighteens to drink alcohol at all in public in the UK, even with parental permission, with the exception of sixteen and seventeen year olds being allowed to drink with a full meal away from the bar area. Note that the pub themselves can be criminally prosecuted in this case, so they will enforce the rules.

That is a shame about the hostel.

1. Globe Theatre. Groundling/standing room. Seats are not expensive.
2. British Museum and / or British Library
3. See a show - something that likely won't hit the States. Although the Play That Went Wrong looks great!
4. Walking Tours - these are great and off beat: Walks.com seems to be the best. Different company: The Muggle Tour for Harry Potter sights was great if into the books but more the films. Tried the latter; will try the former.
5. Fish and chips with mushy peas and curry sauce
6. Pie and mash
7. Indian food