what is the cheapest and fastest way to get from London,UK to Bastogne, Belgium?
I can only tell you how we made that trip. Took the Eurostar to Paris, then Thalys train to Brussels, where we were met by a guide who drove us to Bastogne for a two day tour of the area. I'll bet some of the travel experts on the helpline can suggest a faster and cheaper alternative.
According to the timetables on the German Rail site, you can take the Eurostar from London to Brussels, an IC train to Libramont, and a bus to either Bastogne-Sud or Bastogne-Nord