I think I used to kinda pick it off. If you roll your finger round on it, it balls up and comes off in one go. Hope that makes sense!
Or babywipes.
I can understand why you wouldn't want to use them too early on. I think we went a couple of months before we went to babywipes and never looked back. I have a friend who still uses cotton wool and her oldest daughter is 3!

Don't fancy changing those nappies!