Provided their notice is valid, you are able to go to court to repossess based on their notice (which is not something they can cancel).
If their right to reside in the UK has now come to an end, the home office may assist in removing them.
Giving notice and then not moving out is "holding over" and the tenant is liable for a payment equal to double the rent (although it can't be described as rent).
Have a google.
When I post, I am expressing an opinion - feel free to disagree, I have been wrong before.
Please don't act on my suggestions without checking with a grown-up (ideally some kind of expert).