Just for clarity.
You can only take your pet across the channel if the animal is in a car.

That means:
Via the tunnel it is quick and easy. There are pee spots for pets before you get on the car train and the crossing only takes 45mins

Via Ferry. You pet stays in the car you leave them and wander around the ferry. For the 1.5 hour crossing Calais - Dover this isn't so bad, if you are doing anything longer it can be a lot of stress, especially if there are delays on the crossing.
There is a hovercraft crossing from Cherbourg to Portsmouth which is only 4 hours and you may take pets onboard. Same rules as above.
You can not take your pet on as a foot passenger

You cannot cross the channel with an animal on a passenger train.