@Tiredmaturestudent I'm a single parent and am part of a local group so know lots of others with a variety of jobs/shifts. Many are health care workers too. I am, but now work in a related role and not directly health care.
These are not high income people, for example in the group we have a midwife, two nurses, a mental health crisis worker. All those I mentioned have shift work that includes weekends and nights. Only one has family help. Only one has an ex-partner who is involved, in her case he's loving and kind, but due to mental health can't be relied upon for child care.
One has an au pair, one lets out a room to a postgraduate student in exchange for overnight childcare (the student is lovely, working on Phd), and two hire very sensible University students and recent grads to do overnights, for example arriving at 7 pm, doing bedtime routine and sleeping over. On the whole it's worked well for these friends. Of course there's always some extra juggling as a single parent, and I know one struggled with the occasional night sitter who didn't turn up, but she found a solution and she now hires two - they each do 2 nights/week but it means if one can't do it, the other often can.
In your situation, I'd likely try an au pair at least for the first year in the new job since they'll live with you, I think there's less chance of cancelling etc. Then you could when you've built up experience, and if you wanted that bedroom back, you could try hiring 1-2 night baby-sitters. My friends have found lovely people including trainee teachers. It's a good job for many people.
It's a big leap and you can do it! It's hard but it's possible! Good Luck!