My daughter was born in Kuala Belait, Brunei, in 2007. We had to take the Bruneian birth certificate to the High Commission to get a UK version, then get her UK passport, then the residence stamp, (all within 60 days or she'd have been classed as an illegal resident). Slightly complicated by the fact that Mrs Heidhurtin is Belizean by birth, although naturalised UK citizen at the time of birth. Sorted by the BHC who were great.
My daughter's travelled with me many times to the USA using the ESTA process and never had a problem despite putting Kuala Belait as the place of birth. I haven't, however, tried to obtain a copy of her birth certificate, both originals are closely guarded. This thread is a useful pre-warning......