You absolutely MUST take legal advice. What I say below is only on the basis of some bare facts you have outlined, this is quite complicated.
Right if the house was owned jointly as joint tenants by your father and gran, the house now belongs to him. It does not matter what the Will says.
If the house was owned as tenants in common, your father now owns half the house outright the other half "vests" in the personal representatives of your gran (they have the legal title but no beneficial interest). If there is a Will the people your gran appointed as executor are her prs and her half of the house will go to whoever she states under the Will.
If there is no Will, the people entitled to take out the Grant are the people entitled under intestacy - this will be your gran's children (assuming there is no grandad) and their issue if any of the children have died. Her share of the house will be left equally between the beneficiaries and no one person has a right to live in it.
Thus the only time you are going to have a real problem is if Gran has left a Will stating x is to be able to live in the house.
If Gran has done this you will have a situation where the house is owned jointly between your dad and cousin and both want something different to happen. If it can't be resolved, your dad will need to make an application to the Court under the Trusts of Land and Appointment of Trustees Act 1996 (ToLATA). The court will look at all the factors under s15 and in my view WILL not order the cousin be allowed to live there - the likely outcome is the property will be sold. However, this is where your Aunt holds the ace card - she has a potential claim under the Inheritance (Provision for Family and Dependants) Act 1975 for reasonable financial provision for maintenance either as a child of the deceased or as someone being maintained by the deceased. The upshot is that aunty should stay where she is a resist any attempts to kick her out until this is legally resolved.