My personal belief is that without a degree you shouldn't be able to get a teaching post. But that's not the question you ask. By knocking on doors so to speak you can get a school to take you without a degree, in China and Mongolia both. The rub is that you will have to produce your own visa/RP. If you are married to a Chinese person I think you'd be home and dry. Of course you would have to accept the risk of being ejected from the country if the authorities discover. It will be best to keep your own council if you do follow that path.

Does anyone know if Wall Street English hires teachers with some university credits albeit no complete degree?

Or other schools? Specifically in China or in Mongolia. Could I just show up and get work in Mongolia?

Probably. They are one of the companies always hiring, in every country. Are you sure you want to work there though? Talked to some people who have worked there. It's a weird school.

Probably.
You also may be able to get a job as a doctor, just tell them you have a medical degree. China is in particular need of pediatric surgeons.
I've heard of two Australians who are now pilots in China, scammed their flight certifications.
Lot's of private elementary schools are in need of school bus drivers; just Photoshop a foreign driver's license.
Have fun!