We had this after my partner went back to work after paternal leave when baby was 2 weeks old.
We were both wiped for the first year. Partner didn't do feeds because she was breastfed, but found it really hard to sleep through when she was cluster feeding and cried in the night so would sit up supporting me and making sure I had drinks and snacks.
We just sat down together and discussed what we could do to help each other.
It was things like him taking baby for half an hour before he went to work so I could have a lie down or a coffee before I started the day, agreeing to let housework slip so I could sleep when baby slept, making sure he took her for half an hour in the evening so I could shower, stocking up my side table with bottled drinks and packaged snacks when he was going to the shop
Then for him things like throwing a sandwich in the fridge while I was prepping dinner (takes 2 minutes but makes a big difference in the morning before work and allows him to take baby while I had coffee), making breakfast- even just cereal in a bowl and a jug of milk on the table - when I was finished the 6am feed, or sitting down with him in the evenings and just taking time for a drink and a chat
Then for both of us as a team - he got a robovac so we didn't need to vacuum anywhere near as much, we would tag team, so I would flick the kettle switch or throw the food in the oven while he held her and then usually stop to feed and he would dish up or pour the water and milk into the tea cups
It really does help when you start to work together as a team and communicate. You both need time alone and together while also taking care of babies needs.
We felt like we nailed that first year in the end. It definitely brought us closer despite feeling knackered constantly, I think we both learned we had each others backs.