With 21 days… London>Paris>Brussels>Amsterdam = Train. Munich = Fly, unless you really like trains and/or will get something like a 4 country rail pass. Check Railsaver.com for recommended pass, point-to-point tickets, or combination.

Otherwise, from UK to Paris, see Eurostar.com. For Paris to Brussels, and Brussels to Amsterdam, see the rail site links posted at top of this forum. Tip: check both originating and ending country rail sites, as they sometimes sell seats on the same trains for less than each other. So you’d check the French and Belgian rail sites for Par>Bru, and the Belgian and Dutch rail sites for Bru>Ams.

For flights to Munich, see Skyscanner.net and Momondo.com. Momondo is interesting in that, although it is primarily a flight search engine, the shorter routes that are better-served by train (such as your Lon,Par,Bru,Ams) will sometimes turn up train tickets instead of flights. Worth checking out for this itinerary.

My rules — which pretty much match Don’s suggestion…

Train ride less than 5 hours, take the train — it will be at least as fast as flying and a lot less hectic and more comfortable

Train ride 5 to 7 hours — whichever is cheaper, adding getting to/from airport, any taxes not included in the fare and baggage fees to the air fare.

Train ride over 7 hours — fly unless it’s prohibitively expensive. Look for an overnight train as option 2.

you can take train but flying is more faster and time saver