Due to a break up the property will need to be sold so that they can receive half the equity they are entitled to, the property isn't a joint ownership. Is it possible for someone else to buy their half of the property, the mortgage will then be in joint names with equal rights and pay the other person their half?

If a property is owned by just one party, they can sell 1/2 of it, if they wish, but they will need a new mortgage in both names.

I see no way to force them to do this unless they want to, It may be easier to have the current owner sign papers allowing the other person a lien on the property that has to be paid at the sale of the property