She is asking to stay off until September though… and realistically if OP thinks she might be “starting to look at places” now, no way will her child have a place and be settled in a nursery within the next four weeks. Even if she is deluding herself and honestly believes it might happen, it won’t.
If she was saying “ExDH has completely screwed me over, my DM can watch my child two days a week and I’m desperately looking for a childminder to cover the other days while we are on the waiting list for a nursery”, then ok maybe. But “I’m due back to work this week, and I’ve just started thinking about looking around a few nurseries” is cloud cuckoo land.
This doesn’t reflect the information OP has posted at all. OP has vaguely mentioned September as the date OP thinks nurseries will start having spaces. But they aren’t the only sources of childcare and the new mum hasn’t said she wants to be off that long. OP has described her as desperate to get back.
There are no details on what she’s doing but no indication in OP’s posts that she’s been lackadaisical about the need for childcare.
Even if waiting for September was the plan, hiring and training someone in many jobs would mean that there wouldn’t be much difference in hiring someone new and waiting for September in terms of having someone effective in place. Of course this job may not be complex, so maybe it isn’t the case, but when people leave my team it’s always 6-10 months before we can really count on a replacement making a significant difference to workload. So it could still make business sense especially since OP seems to think very highly of the mum’s work.