It's fair enough you doing the drop off/pick up since you pass the Nursery anyway - however, he should then be helping more in the mornings and at night after Nursery.
You say he gets him dressed but what about everything else - teeth, waking him up, breakfast if he has any etc?
Same at night, he should be spending time with him, dinner, doing his bath/bed etc so you can get some rest before the night wakings.
You say he's not got a commute, does he WFH? If so, could he have dinner going in the slow cooker or similar so that when you get back from the nursery run there's dinner ready for you? Then take over with son once you've all eaten.
When you say sharing nights didn't work - was he not getting up or was it that he couldn't settle him or you couldn't sleep anyway? How often is he waking in the night?
My DP works offshore so when he's off, he's on his time off if you get me (2 weeks away, 2 weeks home) and I'm still on Maternity, we take a night each with our Son, he's in his own room so I wake partner up when I hear him stir and he goes through on his nights. I do have to wake him up because he doesn't wake up to the sound on the monitor but he gets up and goes straight through. It took me a while to be able to actually get back to sleep while he was feeding him as I was so used to doing all the nights myself until 6 months but even just getting to rest in bed without having to get up helped