Thanks for your question. I will try to help.
First of all, if you and you partner presently hold the house jointly (as joint tenants) then each person's share would pass to the other upon death regardless of any directon made in any 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):-
You will then hold you interests as tenants in common, meaning that your respective shares will pass according to their wills or under the intestacy rules. Your partner need not sign the form provided you follow the instructions.
Unless there is a declaration of trust (for which you must hold your property as tenants in common) then there is a presumption that each party retains a 50% interest in the equity in the property .A declaration of trust (had you executed one) would state that you hold your interests in the properties in specified percentage shares (or contributed differing deposit and therefore should receive different amounts from the sale) and it could also say that each person only retains their percentage interest in the property subject to equal contributions to the relevant outgoings for the property
You may conceivably have a claim against the solicitors who acted on your behalf if they did not advise you that you could execute a declaration of trust outlining this.
It is possible to claim greater than 50% of the equity in the property by litigating under the Trusts of Land and Appointment of Trustees Act if it can be shown that you intended to keep your finances separate and it would be inequitable for a 50% interest to be enforce, but this is a very specialised area of law and the legal fees are very expensive. You should ask your solicitor if they are an expert in this and why (if they are) they consider that you would not have a claim.
The contribution of a greater deposit is a fairly good starting points for a claim under the act and I would expect you to be on the front foot in negotiations if a larger settlement is not agreed at the outset.
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