Yes, and maybe.
Yes, it is technically possible to run this train using existing trains and track and border security systems. It would take over five hours until HS2 is built, so an overnight train is probably more attractive. Also, at night the UK west coast main line is less congested. On the French side the train would ‘slot into’ an existing service pattern between Lille and Paris. So why then only maybe?
The key is to find a train operating company that would want to take the risk of possibly losing money by running this service, though the forecast is profitable. International passenger services do not attract UK government franchise payments. Currently in the UK only London has international passenger trains, yet night trains could run to other cities.
On the freight side there is a wide variety of international trains every week passing through the Channel Tunnel, but it is easier to organise trains to the Continent carrying goods than passengers.
I started researching a possible Manchester – Lille – Paris passenger service in 2008, and have written the results in a small book (free) in case anyone is interested, especially a train operating company. Some companies have considered the idea in confidence, and while they did not go forward for various internal reasons, I’m hopeful that one day a company will see the opportunity as worthwhile. Reference:
Night Trains from Manchester to Paris: an outline business case Tony Baldwinson (2012), free as pdf and in Google Books preview. Paperback copies on request.