If you are not married, and the house is in his sole name, the assumption will be that he owns the house outright. If you want to prove that it is otherwise, it will be up to you to give reasons why this should be so.
Normally this is done by contributions to the deposit or helping to finance improvements. Contributions towards household expenses do not count. Contributions towards the mortgage could be, but it is arguable that, if you had the benefit of occupation, it is not unreasonable that you should contribute to the cost.
It is possible that the '' will '' could be used to establish a common intention to share the house, but you would have prove the existence of this document. I''m not all that sure whether it would cut much ice.
The bottom line is that you might have grounds for a claim to a share, but the outcome is hard to predict and in my view the outcome is questionable. However I am not a specialist in this field, and you should most certainly take legal advice before you go down this road.
A more pragmatic solution is that he gives you some money to leave, perhaps enough for a rental deposit and the purchase of essential furniture.