There is at least the possibility of a criminal trial for misconduct in public office.
The Crown Prosecution Service (CPS) publishes the criteria to make out the offence as
The offence is committed when:
- a public officer acting as such;
- wilfully neglects to perform his duty and/or wilfully misconducts himself;
- to such a degree as to amount to an abuse of the public's trust in the office holder;
- without reasonable excuse or justification.
There is a high probability that all four elements could be substantiated if a Prime Minister wilfully neglected to do what the law required.
Misconduct in public office is an offence at common law triable only on indictment. It carries a maximum sentence of life imprisonment. It is an offence confined to those who are public office holders and is committed when the office holder acts (or fails to act) in a way that constitutes a breach of the duties of that office.
The Court of Appeal has made it clear that the offence should be strictly confined. It can raise complex and sometimes sensitive issues. Area Prosecutors should therefore consider seeking the advice of the Director’s Legal Advisor to resolve any uncertainty as to whether it would be appropriate to bring a prosecution for such an offence.
This would be, in effect, an impeachment. As the political ramifications of the inevitable vote of no confidence succeeding (resulting in the fall of the Government) would proceed faster than the legal process, it's doubtful that bringing a prosecution would be "appropriate" by the time the decision was made. But it is at least a theoretical possibility.