I agree with both responses - take the train.
The route to the train is slightly more complex than mentioned because trains from Windsor or Hampton Court Palace don't go to St Pancras - a tube/bus/taxi/bicycle journey between two London train stations is required.
From Windsor there are two routes by train to London - a slower, less convenient route by South West from Windsor and Eton Riverside Station into London Waterloo, and then Waterloo to St Pancras by tube (Northern Line to Leicester Square change to Piccadilly Line to Kings Cross/St Pancras, or Northern Line to Warren Street change to Victoria Line to Kings Cross/St Pancras) and then a quick walk to the Eurostar check in desk. In Brussels it is a very easy change to a train to Brugge/Bruges.
The better train from Windsor is the Great Western service from Windsor and Eton Central Station directly across from the castle, into London Paddington (quick easy change a Slough, the train between Slough and Windsor is a tiny branch line service which just goes one stop around the corner and shuttles back and forth constantly) and from Paddington it is easy to get get to Kings Cross/St Pancras by Circle Line or Hammersmith and City Line which share the same track.
From Hampton Court Palace it is very easy too, just walk across the river and the station is right there. That train is South West and goes into Waterloo - then follow the Waterloo directions above.