There is no way you can apply for the birth certificate of another country. You can however, get a permanent residency of citizenship of another country. You have mentioned that your husband has problems getting his birth certificate from Nigeria. I believe Nigeria is a very Bureaucratic country so things do get difficult sometimes. However, the best way for him would be to contact his relatives or family in Nigeria who can get him his birth certificate through local channels.
My mother's father was in the military when she was born but in Oxford Germany, who can we get a copy of her birth certificate