New rules recently introduced mean that anyone who has to deal with "sensitive" contacts with the USA has to have a British passport. Many dual nationals and Republic of Ireland nationals, who are already serving, have been told that they will have to get a British passport. Indeed a new fast track service has been set up, a weekend in Paris at the taxpayer's expense (because you can only apply for nationality outside the UK).
The bottom line is that, especially for aircrew, you will probably have to have a British passport. But why not go to the local British Embassy and ask?