The Channel Tunnel has security checks because it's a 30km undersea tunnel, and separates an island country with a different approach to weapons to its neighbour. There are no security checks on any other trains in the UK.
It's also not necessary (in Europe) to buy tickets any differently when crossing borders. Advance-purchase tickets are used for long-distance high-speed trains where they don't want people sitting, or wish to spread the demand throughout the day to avoid crowding — that applies whether or not a border is crossed.
I can buy a ticket (paper or electronic) moments before the train from Copenhagen to Malmö leaves, since it's a medium-distance regional train without reserved seating.
It's also not necessary (in Europe) to buy tickets any differently when crossing borders. Advance-purchase tickets are used for long-distance high-speed trains where they don't want people sitting, or wish to spread the demand throughout the day to avoid crowding — that applies whether or not a border is crossed.
I can buy a ticket (paper or electronic) moments before the train from Copenhagen to Malmö leaves, since it's a medium-distance regional train without reserved seating.