In years gone past there used to be 50 seats booked 3 times a week on BA flights to Washington, which people could apply for using the standard indulgence method.
The same was true for BA flights to Hong Kong (well I did say it was years ago).
While I can fully appreciate that the HK route is no more, given the amount of military personnel visiting Washington on a regular basis, embassy staff, Pentagon visits, etc, is there currently anything in place for getting UK military personnel to the USA?