If you are staying in London with teens, there are good places to visit, but a lot of depends on your teens’ tastes.
Go to the Tower of London and take the Beefeater tour (yeoman warders are what they’re called). Try to go first thing in morning before the crowds.
Take them to Camden Market for shopping. Other shops they’ll like are Top Shop and H&M. The London Eye is fun.Consider a day trip outside London also. London Walks has a lot of interesting walks and some day trips. A lot of people like the Jack the Ripper Walk.
If your teenagers would love Camden for shopping, think they might find Madame Tussauds a bit boring at 16. You can ride on the Docklands light railways to Island Gardens, get off here (great views of Greenwich) and then walk through the foot tunnel under the Thames, you can hear the tube trains in the adjacent tunnels and it sounds like there right behind you. Give the museums at Greenwich a miss but climb up to the Royal Observatory and stand on the meridian line.
They might like the National Portrait Gallery too. It’s a great history of London through it’s people and even includes up to date paintings and sculpture of some living Brits.
Take them to the Old Bailey (court house) to watch a trial. You can visit Tate Modern – the big gallery on the south bank – free to see the permanent exhibition – big and impressive with lots of shock value, cool shops and cafe.
St. Pauls Cathedral is well worth a visit – climb all the way up the Dome and look out across London.