Z visa for a non-native English teacher

Non-native speakers from European countries, or (for want of a better description) "white-skinned" are in as much high demand as native speakers. The only difference is that the salaries for native speakers are much higher than that of non-native speakers. I don't foresee much problems for you getting your Z type, if any at all.

Regardless of the schools being in the North or South, employers either want native speakers or non-native speakers, so they will process the work permit for who they ask for. However, some schools don't have the license to hire foreign teachers at all, in which case they just try to hire teachers who already have a work Z.

If the school has already accepted you, and you're not in China yet, you shouldn't worry, they will do all the hard lifting on your behalf.