Firstly, if you presently hold your interests in the house jointly (as joint tenants) then each person's share woulld pass to the other upon death regardless of any director made in the Will.
If this is not what you want then you should sever the joint tenancy by using Form SEV from the Land Registry (you will have to send it to them and if you have any questions about completing the form you should call their customer service number - they are very helpful):-
Your ex-partner need not sign the form provided you follow the instructions. You will then hold your shares as what is known as "tenant in common" and each person's share would pass according to the person's Will or under the intestacy rules upon death.
If you both named on the title to the property then you both have a right to occupy the property that neither can deny the other. This remains the case unless there is a Court order to the contrary made pursuant either to civil or criminal proceedings. He could in theory move back in which would obviously be hugely uncomfortable if your new partner is in the property.
Before your new partner moves in you will have to speak with your lender and ask them for their standard form Deed of Consent and Postponement. Your new partner will have to sign this at a local frim of solicitors, it confirms that he does not have and will not acquire an interest in the property in priority to the lender's charge. It is a condition of your mortgage that any occupier signs this.
In terms of forcing him to sell then you can as a solicitor to write to him advising that you intend to apply for an order for sale if he cannot agree to sell the property and be paid a sum in respect of his equity. If he does not comply then the only option you have is to apply for an order for sale of the property, but if he is still paying half the mortgage then your chances of obtaining an order are a bit diminished because of this.
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