Sorry, I am seriously unqualified to give advice about Visas.
The Eurostar is great, especially if it's easy for you to get to Kings Cross.
Paris would also be a place to go, but I prefer Bruge. If you like the beach, you can book a hotel in Ostend, which is cheaper than Bruge but has a great beach.
Check the Guardian for good travel deals.
Do you drive? Another option would be to take the tunnel and go to Bolougne Sur Mer. It's a 4 hour trip from London, but it's France and there's a great aquarium.
If you live on the South Coast, there's always the Newhaven to Dieppe ferry. Dieppe is a pretty town with a busy market on Saturday