We have a constitution - it just isn't codified [1]. Therefore doing so would just mean having a set of written laws that effectively make public healthcare part of the UK political structure.
Labour (under Jeremy Corbyn) shares a lot of blame for this situation. Why? In the last election, Labour refused to take a stance on remaining in the UK [1]. To be fair, there was also a massive, coordinated and successful smear campaign against Corbyn. Brexit won by a narrow majority (52-48) and there was no party to vote for to vote against it in the election.
The Conservatives went from having a narrow majority (and requiring DUP support for that) to having a massive majority not seen in 40+ years.
And now the tories are imploding it looks like we are going to get a massive Labour majority at the next election. And that probably isn't good either. Especially given that Labour prefer to spend all their energy fighting each other.
I think the antiquated first-past-the-post system is at least partly to blame for the mess that is UK politics.