The hospitals here use cloth.
(I ended up using some diapers for me when I ran out of the big pads they gave me actually)!
So we found the cloth easy (actually my husband did all the changes while I rested for that 5 days in hospital
). They even had those big safety pins. I think they were unbleached cotton ones but quite small, perfect for newborns.
That said, For me personally, I couldn't move much so there was no chance of me changing any diapers let alone figuring out the cloth one at the beginning.
However, if all they have is disposable ones and you're not sure of the cloth then there are some options like the unbleached 7th generation? ones, or even a g-diaper one (those your can flush or they biodegrade but I don't know how those ones would work with the stump: we used those at the beginning months a few times going out when the stump fell off). Maybe the 7th generation ones have newborn sizes?
At hospital , we folded the cloth one over so it wouldn't hit the cord stump.
So, whatever feels best for you. If you're worried about not being able to wash them right away then go for the gentler alternatives of non-bleach ones or biodegradable ones.
We did use disposables first few weeks: baby ended up with a horrible rash. The dr. said it was from using the brand we used! (non-bleach ones, etc.) but I am not so sure. Baby gets a rash if she is wet for 1 second! and the rash was controlled using cloth diapers.
PS: Do all your hospitals use disposables? Isn't that expensive for them? Just curious!