Amsterdam, Paris, and London

It would be easy to take a train to Bruges, then Paris and finally to London.

Thalys is your train service for Amsterdam > Brussels > Paris
https://m.thalys.com

Eurostar is your train service for Paris > London
https://m.eurostar.com/#/homepage

Amsterdam-Bruges - take a fast Thalys train (necessary to book in advance at the address above) or a slower Inter-City train (don't book in advance, it's cheaper than a Thalys ticket booked on the day, but more expensive than an advance Thalys ticket) to Antwerp and then a domestic Belgian train to Bruges. There are no discounts or seat reservations on domestic Belgian trains, so book at the time. It may be cheaper to get a through ticket for the whole journey on Thalys in advance.

Bruges-Paris - take a domestic Belgian train to Brussels Midi and then Thalys to Paris. Book in advance for the whole trip with Thalys.

Paris-London - book ASAP through Eurostar for the best prices.

This is the easiest route with no backtracking.

For us, summer in Paris was wonderful with perfect weather, and London was hot! So, you just never know. Try to get AC wherever you stay just in case you need it.

I have had a/c in every hotel or apartment I have stayed in including Paris. The only exception was Lucerne but they did provide a fan.

PS: Note that when changing for Bruges in Antwerp, the local spelling is "Brugge".

Thanks for the response. You've all been exceptionally helpful.