British Citizenship for my wife (We're both American)

[panthers75]

I have a follow-up question.

As I'd stated, my wife's father was born in Belfast in 1945. We have a copy of his birth certificate (official stamped copy). However, when he was born, his name was Harry Harvey Smith. After he moved to the US with his mother (British) and father (American) when he was 2-3 years old, his father passed away (a few years after they moved to the US). Herein lies my question...

After his father passed away, his mother remarried a man named Joseph Wilson. My wife's father was not adopted, but had his name changed from Harry Harvey Smith to Harry Joseph Wilson. On my wife's birth certificate, her father's country of origin was listed as "Ireland," despite Belfast being in Northern Irelend (UK). We've tried to track down the name change documentation by going to the counties where we suspect it occured, as well as asked the state where we believe the name change had occured. Unfortunately, we've been unable to find this documentation.

My question is - can she still get a British Passport? Our main goal is to have a document proving her British Citizenship, hence the passport. Sorry for all of the confusion, but these are the facts.