The gates close 30 min before departure of the Eurostar, they don’t wait for other trains afaik.
From Hamburg to Brussels you only have one connection, in Cologne. This is fairly reasonable (45 min), and you have a bit of padding in Brussels (you can handle a delay of 45 min).
If you’re unsure about the connection. you can take a train (maybe two) earlier in Hamburg and arrive in Brussels much earlier, have dinner there, and do a bit of sightseeing before going to the Eurostar terminal. I’lld suggest you do this, leave at +/- 8h45 or 10h45 in Hamburg and aim for the last Eurostar of the day from Brussels.
If you do get “deutschebahned” when taking trains two or even four hours earlier, you have plenty of buffer of two hours for the next ICE getting you in Brussels or continue with regional trains across the Belgian border to avoid pricey Eurostar reservations between Cologne and Brussels.
The route with regional trains in case the ICE has troubles is advisable (RE from Cologne to Aachen, in Aachen you can catch the ICE to Brussels or a local train to Welkenraedt, and from Welkenraedt via Belgian IC-services directly to Brussels.
They have new trains on the connection Frankfurt-Cologne-Brussels now, and it’s much more reliable than 1-2 years ago. Last month they only had two times a delay of 60 minutes and no cancellations, which would make you miss the Eurostar to London. In 94% of the cases, you would’ve made the connection :) That’s quite high.
Eurostar isn’t a train you want to miss, so just leave earlier from Hamburg if possible.
