Quick summary as I could write pages:
So we assume the house and mortgage are wholly in your name - on the face of it he has no claim on the house. That is the starting point in law.
However he would have to claim that he had an interest in the house under a constructive or resulting trust. This is very difficult to prove.
For evidence of a trust you need to consider whose name all the bills for the house were in, who paid for all the upkeep and maintenance and who paid for any capital expenditure on the house, such as the extension. This would be relevant for the time he has lived in the house.
Consider: When he paid for the extension how did he pay? Did he pay you into your bank account? or did he pay the builder directly? how much input (financial and decisions) did he have over the extension?
Other factors to consider are whether he paid a sum to you as rent and the overall financial set up.
Your and his intention regarding ownership of the house is very relevant, do you have any emails, letters or texts showing your intention?
There is another legal situation you should be aware of - under the Married Women's Property Act and its subsequent variations, a engaged couple can have their property considered as if they were married, and therefore a fiance could have a claim. However the courts have precluded claims where a couple has been engaged for over two or three years and the marriage has never happened. So if your exDP tries this approach in the circumstances he will fail.
Practically speaking will your exDP actually go to court to claim an interest? unlikely it would seem. For the amount of £18000 it is hardly worth going, especially considering the effort and the fees.
What do you think is fair? Why don't you sit down and work out your financial contributions to your life together and the children objectively - and if you can show in all fairness that you have paid for thousands more than him then maybe he is being unfair asking.
I guess someone has told him he can get his money back, and he is out for revenge after a nasty break up, but it is rarely that simple to prove his entitlement.
I suggest you gather together all the evidence - both what he might try and for you, and a summary of your financial outlays; This makes it easier for a lawyer to advise you. Cases involving cohabitation and financial contributions are really complicated, and impossible to advise without enormous amounts of evidence.
I would say legally he has a weak claim from what you said, but practically you could make a small settlement offer, and stick with that at the moment. I also think you should point out to him his financial obligations past and future for the DC, and give him a 'bill' for that. It might focus his mind.
Do post my mistake if I have missed something really important, I just skim read through.