To my mind, at the moment, you have no agreement - or so it seems - and therefore he can''t take you to Court if there is no agreement.
You see, the house is an important part of any financial agreement, but it isn''t by any means the only one. There are questions of spousal/child maintenance. There could be issues of pensions or other assets.
Put crudely, you can do one of two things.
Either you come to a financial agreement with your x2b, or you have to go to Court and the Court will decide for you. For obvious reasons it is better if you negotiate.
The child support is not affected by your new relationship. Spousal maintenance almost certainly will be affected by it. But you should keep some SM, even at a nominal figure. Cohabitations can break down, and you may need to be able to revive it if that happens.
You should take to your solicitor all your financial details and those of your x2b as (s)he will not be able to get very far without the information.