Sorry not sure about transfer arrangements to your accomodation. The tube is generally safe during the day, I do know of a few bag snatches but in general I travel on it and have been fine.
Wouldn't recommend tube travel after about 8pm though, take a black cab.
Do the main sites by taking a tube to Piccadilly circus, walk down from there to trafalger square to see Nelson's Column, cross the road and walk down the mall to Buckingham Palace and then back through the park to see the changing of the guard at Horseguards (usually 11am). Walk through the Admiralty and turn right past the cenotaph and down to Westminster Abbey/Houses of Parliament and Big Ben and then go on the London Eye. This walk will literally only take about 90 minutes.
If your children are interested in the War there is a great exhibition under the Admiralty called the Cabinet War Rooms which is basically the underground rooms used by Churchill in World War two.
A bus tour is a good way to get your bearings of the main sites and is fairly cheap about £12 per adult and you can hop on and off as many times as you like.
Another excellent trip is to go to the Tower of London to see the crown jewels (teenagers will probably love the gore of seeing where various people had their heads chopped off!!) Then jump on a thames cruiser and get off at Greenwich to see the maritime museum and the line of Greenwich Mean Time. Back on the cruiser and cross tower bridge and have a look around a British battleship 'HMS Belfast'.
Lots of other fantastic things to see and great shopping too but I would definitely give Madame Tussauds a miss. None of the models actually look very much like anyone and most teenagers will be bored within minutes.
The main museums are great (and free). At the top of Trafalger Square is the National Gallery (for art lovers). The Natural History Museum has a number of dinosaur skeletons, the Victoria and Albert Museum (furniture, social history, toys) and the Science Museum are also really good. Finally the Imperial War Museum is excellent (has the smallest boat to cross to Dunkirk in World War Two, early tanks etc.) but DON'T take the tube here the area in which the IWM is situated can be highly intimidating if you don't know London.
Any more requests ask away! Hope this has been of help. Enjoy your trip. Which month are you visiting?