Thank you for your question.
In the first instance your daughter should check to make sure that she and her ex-partner hold their respective interests as tenants-in-common. Were they to hold their interests as joint tenants and your daughter passed away then her interest in the property would automatically pass to her ex-partner irrespective of any Will she has (or the intestacy rules).
If held as joint tenants you should sever the joint tenancy by Form SEV immediately (downloadable from the Land Registry website). If you have any questions you should call the LR; they are very helpful.
If he cannot get a mortgage to finance a transfer of equity and remortgage then you are left in a stalemate.
They should try as hard as possible to come to an agreement to sell the property and the proportion of any proceeds of sale realised, this will avoid unnecessary expense and you should impress this logic upon your daughter's ex-partner in the hope that he will be come around.
If they cannot agree then your daughter will have to apply to Court for a order for sale. The split will be a notional 50/50 but parties can make representations to depart from this given their particular circumstances. It will likely take 2-3 months and you should speak to a local solicitor about acting on your daughter's behalf. These application are seldom refused.
A difficult but unfortunately common situation, I sympathise.
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