Yes agree with Carang, if you wife is here on a dependent visa then it doesn't make sense why she didn't get a HKID in the first place, as on a temporary visa you keep having to exit HK to renew the visa (I came on a dependent visa with my husband from Australia) and that would have been a TOTAL hassle, and it seems as if you guys have been here for some months to half a year from your post.
Irrespective, I think you've had quite a bit of great advice from the ladies and men on this forum i.e. to ask work to write a letter in support of your continuing employment or ask them to renew the contract early ect.
I wouldn't stress out, once your wife gets the HKID she can register in the hospital within her geographical area and it doesn't matter if they are full or otherwise, they will HAVE to take her onboard no matter how far along she is. In the mean time, the antenatal care probably needs to be done in the private (the private don't care if you have a HKID or not - so long as you can pay their fees --> CAPITALISM pure and simple).
I would try and sort it out ASAP though as there are different rules for Mainland women when they marry locals (and I'm not sure if this applies to Mainland ladies who marry non HK residents). Carang is right in that Mainland ladies that marry local HK men are not entitled to live in HK with their husbands for around 5 years, they are issued with visas to visit regularly I think but they can't stay here permanently and certainly can't work here. Moreover, they are NOT entitled to a HKID.