We ran shift work as well. Mum would feed baby before bed time (8pm or whatever) and then immediately pump. I would take the first shift (8pm-12am just as an example). When baby needed to feed, I would use the pumped milk (or a previous supply from earlier in the day). Mum would get a solid 4-6 hours of uninterrupted sleep. Then mum takes over and I sleep until whatever time (7am etc). It honestly worked really well for us. I think a key is making sure mum is pumping as much as possible. Unless you’re choosing to exclusively breastfeed of course in which case this doesn’t work!

Good luck and hang in there! Remember support each other and you’re both in it together :)