my 18 month old DS has been the same - in his case, there were a lot of ear infections causing the wakings, but even once they were pretty much cleared, he was waking and wanting to party at 3 am. I had words with him more than once. We still have him in his cot in our room (no space for him to move yet), and did used to bring him into our bed immediately he woke, and then I'd feed him, then we'd lose our tempers with this wriggling loon in bed with us - it was getting really bad. He is also teething, but quite frankly I don't care at the moment - I have 2 wisdom teeth coming in and they aren't enough to stop me sleeping
We've just resorted to a mixture of controlled crying and PUPD - it's been 10 days or so now, and he seems to finally understand that once he is in his cot, he bloody well stays there until the morning.
every time he whimpers, we lie him down in his cot, say 'sleep' and pat him on the chest, then walk away. Someone on here recommended this to me, and although DH and I were totally knackered when we started it, I decided enough was enough, and we've gone for being TOTALLY consistent - same response, every single time DS stirs. We do this at bed time - he goes down in his cot, we leave the room, he yells for a bit, we return, lie him back down, leave again. we're not doing the proper CC timings as he gets too distressed after more than a minute or two, but after 3 or 4 returns last night he was asleep. the first night took 3 hours at bed time, and again from 3 am until 6 am. Work was fun that day... second night 2.5 hours, third night he had a fever, 4th night he was up every hour, etc etc finally last night he took 10 minutes to go to sleep
, and although he woke a few times in the night, went back down each time with no fuss. He doesn't have a night feed anymore - no matter how much he claws at me for one. We give him water every time he wakes instead (and he has a feed before bed and first thing in the morning). If there's a way you can start this training just before your ex has them for a day, maybe you could get some sleep during the day too, although I appreciate this may not be totally straightforward.
I will also say, I feel like a very cruel mummy for doing this, but, god, I also feel like a rested human again, so I think it's totally worth it.
good luck, and I'm also here for hand holding if you need it!