"How old is your unborn baby?" is not grammatically wrong, but is not the way people usually ask the question. We normally count age from birth, not conception -- I presume because we know exactly when a baby is born, but we rarely know for sure when he was conceived -- so to be consistent an unborn baby's age would be a negative number!
The common way to ask the question is, "How far along are you?" or "How far along is your baby?" A woman will often say, "I am three months pregnant". But then we also say "the baby is three months old" or whatever number.
Doctors will say that a baby is "three months gestation". Sometimes they talk about age since the mother's last menstrual period, as in, "the baby is three months LMP". LMP is also a date that the mother is likely to know. It's usually a couple of weeks before conception, but close enough. Medical researchers talk about "embryonic age", which may be abbreviated to simply "age". But these terms are rarely used in casual conversation.