Hi, I am a mainland mum with English husband, and I deliverid in HK, my daughter is turning to 2 now. It was same situation we had, but luckily my husband got insurance to covered private hospital, we applied Matilda when I was 6 weeks pregnant, i didnt have my ID card in that time as it takes so long to apply for,they accepted my passport as a application documents, then I got my ID card when I was 3 months pregnant, so it was all went well.
Just thinking May be you can apply for private hospital now incase that you can't get a spot in public hospital when your wife got an ID card, as it will be too late until her due date?
Or delivery in Shenzhen, you can apply a Depandent visa for your child once she/he born, it takes 4-6 weeks.
Hope that helps, You can PM me if your wife needs any one to talk, it must be so stressful for her.
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