This is a muddle:
1. From what you say, you did not receive anything from the estate (it all went to your mother). If so, you cannot be liable personally unless no advert. was placed.
2. Even if an advert. wasn't placed, if the debt is due at all it is arguable that it is due from the estate - i.e. from your father's share of the house which was transferred to your mother. The creditor could make the estate bankrupt in order to recover their money. This could conceivably lead to the house having to be sold to raise the money. (I'm assuming the house was the main asset & that he didn't leave anything else of value.)
3. The solicitor was presumably not an executor. His responsibility therefore depends on what you instructed him to do. If you simply asked him to transfer the house then he won't have any liability, but if he was instructed more generally in relation to the estate he should have advised you of the need for an advert. If he failed to do so, you could complain of his negligence & go to the Legal Ombudsman if necessary.
4. If there was an advert., & it was correctly dealt with in accordance with the law, then you do not have any liability.
5. It is very odd the hospital raised this (apparently) soon after the death. I very much doubt that they would have failed to invoice & chase the debt for 3 years. It could be that they had been chasing it but your father had done nothing about paying it. How did the solicitor find out about it? Why would the hosiptal raise it with them?
6. As you can see, there are potentially a lot of issues here. You should start by finding out about the advert. (or its absence) & getting a proper explanation from the solicitor of why he says you are personally liable.