Some people are happy to have their children live with them long after they become legal adults at 18. But for many reasons, this might become an untenable situation. This could be because your adult child is not contributing anything to the house, or because of disrespect or violent issues. As a parent you are legally entitled to ask your adult child to leave your house.
First of all it's important to get advice from professionals, ideally certified family lawyers, to fully understand your legal position. If you are short on funds, there are public free advice services, such as the Citizen's Advice Bureau.
Your adult children are not legally tenants, but "excluded occupiers." That means you do not need a court order to evict them, and just giving them notice can be sufficient. Keep a record of all communications, so sending a letter might be a good idea.
If the notice period ends and your adult child does not leave, you can take action like changing the locks and putting his or her possessions outside (as long as they don't get damaged). Careful not to damage any of their possessions or use any violence, otherwise you could face legal action yourself.
If your child is over 16 but under 18, you cannot legally evict them. But there are schemes in some areas which will take 16 to 17 year old children off your hands in order to put them into accommodation and employment. Use the "find your local council" link below and call for details.
After the notice has expired, you can get a possession order from the court in order to evict your "excluded occupier," or adult child. Your child might argue that, because they have been living in your house for such a long time, the notice period should be longer. But as long as you have given reasonable notice the court should grant your order.