This could be a bit rambling so I'll try to make it as simple as possible!
DC is due on 13th July, but obviously could come anytime a couple of weeks either side of that.
DH has got a new job but unfortunately needed to give 9 weeks notice to his current employers, meaning that he is due to finish one job on 21st July, and start the next on 26th July.
He gets 2 weeks full pay from his current employers, assuming he's working there when I give birth. If he leaves their employment midway through this 2 weeks, I presume they have no reason to pay the complete amount?
He would also get 2 weeks full pay from his new employers (even if I gave birth the day he started) but I presume they wouldn't have to pay him anything if I gave birth the day before his contract officially started?
I hope this makes sense!