The train trip between London and Paris lasts an impressive 2 hours and 15 minutes nowdays. Eurostar service is the most convenient way of travelling to Paris. The train itsef is silent and comfortable. It even offers power sockets next to each seat for business travellers. During the booking process they even let choose you seat depending on whether your connection is the UK or the European one.
The big but is that Eurostar do not not offer WiFi on-board. It does does offer free access at the St. Pancras International station. You can connect in the waiting area and also on the train until you leave the station behind.
The network SSID is ‘StPancras WiFi’. It is an Open network so make sure that the encryption WEP or else is disabled. The network is provided by ‘Access point only (infrastructure).
I did not detect any network at the Gare du Nord in Paris, so there might be none there.