An experienced nanny would likely have been through this before and would have known to include it - unless you were lucky to have extra generous and conscientious employers before now.
I think you're getting no replies because no one knows what you're asking. Should you quit now or wait? Only you can decide that... Yes, it will be harder to get a job without a license but it is not impossible. I don't know what your cost of living is, how much you have saved, how marketable you are, or what your local market looks like, so I can't say whether it makes sense to stay until you get a license or not.
I will say that you should begin looking for a new job BEFORE you quit. Why not start now and if you find someone willing to hire you without a license, great, go ahead and quit. If not, you will have at least a crappy job to pay the bills until you become a stronger candidate?