As someone who went through the same thing a week and a half ago, I can honestly say that there's no way you can plan for a shoot like that... Believe me, I tried and failed miserably!
I went into the delivery suite beforehand at different times and took meter readings to get an idea about what the light would be light but as soon as my Other Half went into labour, the whole bloody lot went straight out of the window! :bang:
In the end, I went in with the Fuji X10 set on EXR mode and fired away as much as possible whilst trying to be useful to my partner.
I think you did really well and created a lovely memento for the new parents!