Em, we are going to face exactly the same problem as we are moving to another city, where we know no one, when I'm going to be 25 weeks pregnant (later this month). DS1 is 3 1/2 so maybe not quite as much of a challenge as your little one, but still needs looking after. My mom, like yours, is completely unwilling to help, and mother-in-law is completely unable, and no other relatives around near enough to pitch in, and no friends there. Neighbours-to-be are elderly on one side, and students on the other, so no possibility there either.
We have no definite solution, but I'm thinking that we will try to get a "trusted babysitter" going as soon as we can when we move to the new city, so that there is someone used to DS and our house and his routines. Then, we just have to hope that she will be available when the time comes...We also thought of having an au pair for a few months so there would be someone actually living in the house who DS would be familiar with(yes, I know, there are many threads on here about au pairs not having sole charge of small kids but many have a year or more experience and if you can find one of these looking to extend her stay in England for a few months, could be just the ticket). We haven't completely ruled this out, but I think having just moved ourselves we may not be up for having a new person come to live-in full time right away.
Not sure if this helps but just wanted you to know you're not alone in this dilemma! Good luck. I am sure you will find a solution. Sometimes outside help is just the simplest - as long as you are not expecting someone to just pop up on the day itself and slot right in with no problems. And I would really second others on here as far as NOT bringing your little one along to the hospital...