Is it such a surprise that she could have commanded the ship?
After all there is already a precedent:
T'Pol as XO of NX-01 and Kurn as XO of E-D.
Both of them were not Starfleet, but were XOs, so could have (and did) command a Starfleet starship...
You forgot the glorious service of Riker to the Pagh, who was entrusted to defeat the Enterprise.
US Navy and British Navy might share a port for instance and have an agreement between command officers but a British commander isn't getting on a US Navy ship and taking it out on a mission or vice-versa
Britain will not likely become an American state, as Bajor was a potential member of the Federation. If that process is going forward, the incorporation of the Bajoran military personnel would happen.