I'm also a labor nurse and it's our hospital policy not give baths until at least 2 hours after the birth for the reasons mentioned above--mostly temperature instability. We also give parents the option of warming up their baby using skin to skin--usually covering both with warm blankets to assist with that. I've seen really "messy" babies and really "clean" babies, but in general after the initial wipe off, the only way I can tell if a baby has been bathed is by checking the chart or by checking his or her hair.
I also ask every parent if they'd like me to bathe their baby or if they'd like to do it themselves--and I'd say about 20% take me up on it to do it themselves. We also do it in the LDRP room, so there is no separation.

And I never mind if they say they'd like to postpone it or decline to have it done at all. Less work for me!
I'm planning on home births when I'm pregnant, but in general, I think it would depend on how covered the baby is blood, mec, etc. Mostly I think I'd just leave the baby alone--at the very least until we'd breastfed and cuddled for quite sometime.
Oh, and a note on the scrub brush mentioned in an above post--I was skeptical, but I've found that almost all the babies LOVE it. It's a very soft bristled brush with a sponge on one side--great at getting the gunk out of the hair (if you want it out).
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