Great advice from the Man in Seat Sixty-One (fascinating site, great for stirring the imagination). Looking at rail journeys and the advice for Amsterdam - London is to book via a Dutch site instead of via eurostar. So I priced in both, on the exactly the same trains -- a thalys high-speed amsterdam-Brussels, eurostar to London. For 3 of us, eurostar came up with 488 GBP(!), whereas nshispeed quoted 385 EUR (about 345 GBP). That's huge. Again, exact same trains & times. Eurostar, you suck.
Furthermore, eurostar doesn't seem to offer a normal intercity links from amsterdam via their booking site. Why? Because eurostar sucks. nshispeed does, of course, and I see that if I wanted to leave 23 minutes earlier I can get to London at the same time and save another 83 EUR. So I can save over 200 GBP from the eurostar quote. Or, to put it another way, 90 GBP/person vs. 163/person.