You weren't rude at all. If anything, I'd consider it rude if you asked the neighbour's 'permission'. That diminishes the nanny to possession status.
Even if you wanted the nanny to work for you during her usual hours with the neighbour, again, that's the nanny's business, and not something you need to ask her client about. Apart from being unprofessional, it's rude to request a client's permission when we're talking about an independent adult here. Her client has no say over what she chooses to accept for work, and should not be treated as though she does.
If you wanted to give the neighbour a heads-up, you could have brought up in conversation with her the fact that you're looking for a babysitter, and asked her if she thought the nanny would be good? But given that you don't normally converse with the neighbour, and already know the nanny quite well, this would really be nothing more than an obviously-fake heads-up.
If you see the neighbour or happen to speak with her soon, a breezy "wow, isn't [nanny] fantastic! I see why you employ her so much!" makes it clear that you aren't trying to sneak around behind her back, in case it's the type of neighbour who actually does consider the nanny her possession and that you need to let the neighbour know before you use her.