Right, so what do you think you should pay him?
The one thing about not having agreed anything that I suppose goes in your favour, is that there is no trigger point for you to sell. If he wants money from it, it's him that will have to apply to take you to court.
So the first thing I'd do, is do nothing. Or rather - say nothing. Let it be all work on his side.
I stress that I'm not a lawyer, but I would expect if it went to court, any order for sale would be delayed 2 years until your oldest is 18.
Second thing - if by any chance you're over paying your mortgage, stop - not point in paying off more than you need to until you know what is happening.
Don't propose anything, let him do the running - but work out in your own mind what you think is fair.
I might think like this: say you split when the house was worth £100K, and the mortgage was £50K. I might decide that a fair settlement if you'd sorted it out then would have been 70/30. So at that point, the bank owned 50%, you 35% and him 15%. If the house is now worth £200K, you owe him 15% of it - £30K.
There's 100 opinions of course... but at least based on what you think is fair, you need to start planning how you'll buy him out - because cheating arsehole or not, he is owed money.
If he brings it up again, tell him what you'll pay him and when. And if he doesn't like it, leave the legal runaround and cost to him.
Start saving to see a solicitor though - you really should get proper advice.