Don't bother, is all I can say!
We picked up Boy Two from Senate House shortly after 10:00 am and got a bus down to Waterloo Bridge, and then walked down the steps to the South Bank - but not a Christmas Market hut in sight. Frantic Googling not only ate most of my phone's battery but also discovered that there were markets on the Queen's Walk (near the Eye) and at Bankside. As we were near the Queen's Walk, we went there first - but despite the fact that it was 11:00 nothing was open, and there were only about five stalls anyway. Huge disappointment.
So we decided we'd try Bankside. Since it was the Swan Whisperer's birthday, we thought we'd be extravagant and get a river bus down there, but of course we just missed one! Still, we had the tickets, so we wandered about the Eye. Boy Two asked if we could go on it, but we explained that it was very expensive and the kind of thing we reserved for a birthday treat - we did actually wonder about treating the SW, but he thought not. We said we might take him next birthday.... Anyway, we wandered round the souvenir shop, with the caveat that "We're not buying anything today" (the boys have been brought up to respect that, and don't nag, which is wonderful); both Boy Two and I fell in love with stuffed Corgis with Union flag coats on, but resisted temptation!
After which, it was time to go back to the pier, and we duly caught our riverbus to Bankside. We nipped into Tate Modern to use the loos there, and then explored its Christmas market. Also disappointing, although the stalls were open. We had the obligatory mulled wine - a non-alcoholic punch for Boy Two - and the SW and I both had a Bratwurst in a roll. Boy Two said he would prefer a hamburger, so we got him one - the biggest you ever did see, but he ate almost all of it, most of the onions and half the bun before admitting to being "a bit full up now"! So we mopped up and then walked to London Bridge for a bus back to Walthamstow so that he could have a nap if he needed to, which he did. So, I may say, did the Swan Whisperer!
All in all I was very unimpressed with the Christmas markets this year. Time was, they would stretch down between Waterloo and Westminster bridges, practically, and were splendid with gingerbread and other stalls, but not this year.. I believe there is a third around More London, but we didn't go there, and I don't intend to.