There's an important question here that nobody seems to have mentioned yet and that's "Did Dad actually own half of the house anyway?".
You might think that the answer is "Of course he did. He and Mum owned it together" but it's not as straightforward as that. There are TWO ways in which a couple can own a house together:
Firstly, they can own it as 'tenants in common'. If that's the case then both partners each own half of the property and they're free to leave their share in it to whomever they like in their will.
However they can also own it as 'joint tenants' (which, with married couples, is probably far more likely). If that's the case then neither person owns any part of the house but, instead, it's their legal
partnership which owns the whole of the property. Under such circumstances neither partner can leave their share of the property to anyone in their will (since they don't own such a 'share' anyway). When one partner dies the property is automatically owned outright by the surviving partner and anything written in a will (attempting to bequeath a share of the property to someone else) is totally invalid.
https://www.gov.uk/joint-property-ownership/overview
So the first step should be to check the land registry record to see whether you parents were 'tenants in common' or 'joint tenants'. It will cost you just £3 to get a copy of the title register:
https://www.gov.uk/search-property-information-land-registry
If it shows that your parents were joint tenants then your mother already owns the house outright and she's free to do whatever she wants with it. (However she should first complete form DJP, and send it to the Land Registry to get the title register amended:
https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/490206/DJP.pdf )
If it shows that your parents were tenants in common then (unless you can successfully challenge it via the courts, which could be a very expensive process) the terms of your father's will must apply.