If it had been dealt with before your divorce, as it should have been, then the question of where you live would all have been sorted out, one way or another, when you divided your assets between you. However, this house is neither yours nor your ex-husband's, whether or not this alleged future promise was ever made. How did you divorce if you didn't submit all the paperwork, did you do it yourselves or did you have solicitors acting for each of you? - and if the latter, why didn't you bring up then the question of who would be living in the house in the future?
This is going nowhere as it is - take barmaid's informed and legally-qualified advice, and make sure your own solicitor has all the information. We're an on-line forum, our advice might be good but we can't act for you.
Before you start all this, may I ask again - do you have the £13K to hand, to cover the legal expenses? If not, how would you pay it at the end of the case (particularly, if it goes against you, since you might need that sort of money to help get a new place for you and the children).