Every Chinese jurisdiction has a little bit different time frame, but it also depends on the situation. If this is her first Residence Permit coming off a new Z issued outside the country, it often takes much of the 30 day period. If something is being held up, you'd need to find out whether it's at the Labor Bureau side or the PSB side of things.

If employment changing inside China but different provinces and already have a valid RP in location A, then not so much time to get transferred to new RP in location B...maybe couple of weeks.

Faster still is changing employers and on valid RP but in same jurisdiction, maybe a week.

Fastest is not changing employers, just getting new annual RP issuance, can be about 3 days and the passport part maybe just needed for one day then returned to you.

While your wife is passportless, I'm afraid international travel is a no-no, so yes you are screwed. I have heard of problems in the last year, of not being allowed to fly internally in China on the PSB's temporary receipt they give you while they have your passport. You can try. Train travel and hotel registrations aren't usually a problem with temporary PSB paper and photocopies of the relevant passport pages.