I just filed for a birth certificate for my newborn son at the Frederick Health Department this week. I had a home birth with CNM, as well as OB care for the first part of my pregnancy, so I can't say for sure how much difficulty you'll have with an UP/UC. I'll tell you what I do know, though.
In my interactions with WIC, Medical Assistance, and the birth certificate people, I've found that when I say I'm having a home birth I get almost exactly the same response from everyone. It's usually something like "Oh, that's great. Just in case your midwife can't sign the paperwork for the birth, here's the phone number/form/contact person you need." They are conscious of the fact that there are "illegal" midwives who won't sign the birth certificate papers (rendering it basically the same as a UC in their eyes). They will tell you that you need to have a nurse come to your home and interview you, see the baby, possibly do a newborn exam. This is NOT necessary with a CNM, it's not a blanket rule, but if you UCed you probably won't be able to get around it.
I'm not sure if that covers what you'll need to do for UP as well. As the other poster said, I would call the health department. They've been helpful and not hostile. The birth certificate forms, for each required piece of "evidence", list what documentation you need, but also give the option of "other proof acceptable to the health officer". I would inquire as to what that means.