There was a post on here a little while ago where the woman was being asked to provide emergency childcare on the father's contact time. I can't remember the exact situation but it was something to do with him being unwell IIRC. The overwhelming consensus was very much "his contact time, his problem". Very few people agreed it was reasonable for the mum to step in and look after her children because it wasn't "her" time.
If that's the general consensus of opinion, then I'd say it applied in these circumstances too.
FWIW, I think being flexible with your co-parent is always a good idea. I'm very lucky as although my parents divorced when I was 8, they remained on brilliant terms right up to my dad's death, despite the fact it was his infidelity that ended the marriage. That can't have been easy for my mum, but I'm grateful for the fact she put us first, rather than her own hurt feelings.
I would normally say it's fine to ask OP but you've alluded to the fact that it was certain behaviours of his that causes the marriage to break down and for you to move so far away. It really depends what these were. Without knowing more, and I completely understand why you don't want to say, it's impossible to judge whether asking him to step in was reasonable or not.
It's also difficult to say whether it was a reasonable request without knowing how he feels about the fact you've moved on with someone else so quickly, and are having their baby. If he's struggling with this, then perhaps asking wasn't the best idea.
Either way, him having to take emergency leave is quite a steep ask and not really the same as just switching weekends to accommodate a party etc.
I think so much depends on the circumstances of your split, and your ongoing co-parenting relationship. MN normally says if an emergency crops up in your contact time then it's up to you to sort childcare..... If I was co-parenting I'd prefer the children to come to me, but not everyone agrees with that approach.
Maybe you could use the time between now and July to facilitate a relationship between your new partner's family and your DC? Your DP wouldn't have to be at the hospital for that long really, it's not like potentially waiting several days through labour.
Fingers crossed your baby stays put until you're ready, and you avoid any of these difficulties!