I
think that you've now read the information correctly, ILG but it's definitely not 100% clear.
Trying to find a telephone number to get an accurate answer might be difficult as well, The staff answering TfL's phone lines are basically just reading bus and train times off their screens; they probably haven't got a clue about the intricacies of ticketing. You could give them a try on 0343 222 1234. (Lines open 24 hours per day. Charged at the same rate as normal 01 or 02 numbers - which might be free if you've got 'included minutes' on your mobile package or if you've got a landline account with free calls).
You could also try to find an online enquiry form here:
https://tfl.gov.uk/help-and-contact/
However I've just tried myself and I had to go via an 'untrusted connection' warning and indicate that I wanted to make an enquiry about buses, only to end up with a form that demanded the date and time of the incident I was complaining about!
Some hints for the Tube anyway: Even when the trains are really packed, so that in most carriages you're standing with your nose pressed into someone's armpit, the first and last carriages are often only lightly-loaded, so that you can easily get seats. (So always go to the ends of the platforms).
To get to the Southbank Centre (where you might also encounter me browsing the Christmas Market, BTW!) it's often easier to go to Embankment Station than to Waterloo (because it's on the Circle Line and therefore accessible from lots of other places) and then walk over the footbridge across the Thames. Now that most of the Circle Line trains are the new, airy ones (where you can walk between the carriages), it's also a much more pleasant experience to use them than some of the older rolling stock on the Northern Line.
The walk from the Southbank Centre to Covent Garden is easy enough, over Waterloo Bridge. Bistro 1 is on the edge of Covent Garden (in Southampton Street) and extremely popular with many AB members, including me. An excellent two course lunch there costs £9.90:
http://bistro1.co.uk/menus/bistro1-restaurant-lunch-menu.pdf
If you're walking from there to Oxford Street, it's easy enough to detour either via Trafalgar Square (where you'll find the National Gallery and the National Portrait Gallery) or Bloomsbury (for the British Museum). Many of the exhibits in those places would bore the pants of most teenagers but if you're extremely selective you should find something to interest your trio. (In particular, I recommend the Taylor Wessing Photographic Portrait Prize exhibition at the National Portrait Gallery, which is round the back of the National Gallery).
It's still a bit of a hike to get to, and along Oxford street. Theoretically the best way to see the lights is from the top deck of a bus but even getting on a bus can prove to be difficult (because they're so crowded). It can also take over an hour for a bus to get from one end of Oxford Street to the other in the Christmas rush (written from experience!), so it's far quicker to walk, even with the vast volumes of people thronging the pavement. (I love crowds, so Oxford Street, just before Christmas, is one of my favourite places - but I can well understand why others hate it!).
How well do the children already know London? If they're not familiar with the South Kensington museums, then I'm surprised that you've not included them in your itinerary. They're far more interesting than the Oxford Street lights - and they're all free!
http://www.sciencemuseum.org.uk/
http://www.nhm.ac.uk/
http://www.vam.ac.uk/