Whatever your father's stated wishes may have been, since he didn't leave a will the intestacy rules MUST be followed. From what you've written that means that ownership of the house MUST pass, in equal shares, to you and your siblings:
https://www.gov.uk/inherits-someone-dies-without-will
(Of course that doesn't prevent you and your siblings from coming to an arrangement about who lives in the house or whether it should be sold. If they wanted to, they could even sign over their shares of the house to you but, in the first instance, title to the property MUST pass jointly to all of you).
You will need to apply for a 'grant of representation' in order to distribute your father's estate in accordance with the intestacy rules. As he didn't leave a will, the grant will be in the form of 'letters of administration', rather than 'probate'.
Start by finding the contact details for the nearest Probate Registry:
https://courttribunalfinder.service.gov.uk/search/postcode?aol=Probate
They'll provide the forms and can give you quite a bit of advice too.
Once you've collected all of the relevant information, and entered it onto the paperwork, you simply have to attend the Probate Registry's office to swear an oath (or, if you're non-religious, to 'affirm'). Letters of administration can then be issued to you.
Once you've got the letters of administration, transferring title to the property is simply a matter of a bit of form filling. See here:
https://www.gov.uk/government/publications/whole-of-registered-title-assent-as1
You might need a bit of legal assistance at that point but it certainly shouldn't cost anywhere near to £4000! (Shop around!)