Is your wife visa exempt to Canada? If yes, then you shouldn't have a problem except that as a visitor in Canada, she will not be covered by health care and you will have to pay for the birth which could amount to $5-10,000 depending on if there are complications etc. If she is not visa exempt, it is a question if she would be granted a visa to visit with you. Especially if they know she is pregnant. If they have an idea that somebody might want to overstay or might want to have a baby on their buck, they will not grant the visa.
Have you looked into having the baby in her country of nationality? Have you looked into if either the country where you live or her country of nationality will give citizenship if the baby would otherwise be stateless? Some countries have such an exception in their citizenship law.
As far as I know, there are no major provincial differences to having a baby in Canada. I do not know where you get the idea that it will take 14 months to get a passport for the child. Once you have the birth certificate, you could pay a bit more and have a passport in 24 hours. The normal processing time for a passport is 2 weeks. You may be confusing this with the process of getting proof of citizenship. If you had been born in Canada and could pass on citizenship and your baby was born outside Canada, you would have to apply for this proof of citizenship which can easily take 14 months, however, this does not apply to your case.