I was worrying about this in the run up to having DC2 as we have no local family (nearest are 3.5hrs away), of the good friends in the area, two of the families were on a fortnight's holiday around my due date and the other three were also due with DC2 around the same time as I was and DP had to go overseas with work when I was 39+4 and when I was 41+13. Given DC1 was born at 41+15, the latter was a real concern.
DD already did three days a week at nursery and I spoke to them and they said that they always had space in an emergency so she could have gone there during a week day, I also had the number of a couple of the ladies from nursery and any nanny that I knew. I also typed out instructions on how DD "functioned" (eg home from nursery for toast (bread in x cupboard, jam in fridge) & banana, half an hour of Cbeebies (instructions on how to turn on TV etc), bath, into PJs (where to find PJs etc) which I had both in my kitchen and in my "grab bag".
My "grab bag" was with my hospital bags and contained a spare set of house keys, £20 and the instructions on DD. My idea was that, if I went into labour during the day on a Monday - Friday, I could drop her off at nursery with that & whichever worker brought her home, would be able to get into the house and know what to do as well as have some cash for emergencies. Likewise, if I had to call someone on a weekend day, they would be able to go out to the park, had funds for an ice cream and keys to let themselves back in again.
At the same time, I'd also packed DD an overnight bag with her favourite PJs, a night nappy, a change of clothes and had a note stuck to the front of it saying "cuddly" to remind me in my panic to put in her cuddly. I also had the spare bed made up here. This was as I was fairly sure, despite not having asked them previously, that in a "it's 4am and I need to go to hospital now" type situation, one of the neighbours would have either taken DD or come & stayed here, which they did probably depending on what their own childcare situation was. They would also have been left the grab bag.
I also always made sure that the pushchair was kept in the house rather than the back of the car so that whoever it was who looked after her could have got her to/from nursery or taken her to the park or whatever.
As it was, I was booked for an induction at 40+12, my parents arrived at about 4pm on 40+11, I first felt a twinge giving DD tea an hour later, realised it probably was labour when she was in the bath at 6, my mum did stories and put her to bed whilst I was in the bath, DP got home at 7, we left for hospital at 8pm, DC2 arrived at 9.29pm, I was taken up to the post-natal ward just before midnight when DP left, DD woke up to the news she had a baby brother, DP took her to nursery as usual and then did the supermarket shop before coming to see me when visiting hours started at 10.00am, I was discharged an hour or so later & we were home for lunch. DP spent the afternoon on conference calls and went on his business trip the next day whilst my parents stayed. It was all remarkably straightforward, involved none of my back up plans and, having had such a quick & easy labour, other than needing a babysitter from 8pm - midnight whilst I was in hospital, I could have done without my parents (although it was nice to have them around).
It was all very different to DC1's birth which took three days and left me pretty much immobile for two week afterwards.