Joan and Marcie are being driven in an American jeep driven by an American soldier from Cairo to Tunis, westward across North Africa. American troops were not based in Cairo. They landed in Algeria and Morocco. In Cairo, Kay Kyser also refers to an officer having an eagle on his shoulder (US insignia for a colonel) but this could be a liaison officer.
In the medium and close shots, the Australian and British soldiers are wearing US uniforms, with only different head-wear (slouch hat, berets, etc.) to differentiate them from the US soldiers.