Anybody visiting UK (or using inbound/outbound days) and intending to use the East coast line operated by LNER a reservation is strongly advised and at busy times may be compulsory. These are available free of charge at any UK station or can be pre-ordered from the LNER site if you set up an account, During booking the reservation it asks for a ticket reference but the system accepts a 0 and you simply declare you have a valid ticket.
LNER operate from London Kings Cross which is adjacent to the Eurostar St Pancras terminus. Trains go to York, Leeds, Durham Newcastle and Edinburgh (and a few beyond).