With my first UP/UC in NY, her father just went in and got a birth certificate without a problem.Â
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With my second UP/UC, I don't think we really needed to provide proof of pregnancy, but just in case I had a positive pregnancy test from Planned Parenthood, and I took my son in to the pediatrician's office the day after her was born. I could tell the nurses & doctor were observing/studying me to make sure I was the bio mom and not someone that just cut the baby out of a mama's belly. 
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These births were in 2001 & 2002, btw, and in two different cities in NY. I don't know if the laws have changed since then. Hasn't really applied to me.Â
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My friend was able to take a hpt at the WIC office, and they used that as proof of pregnancy for her to get services.Â