at Lux there is an airside route so a separate connections channel but I don’t think it is ever open. I happened to be there just this weekend and t was closed again. Tbh you are better off just going through immigration when you arrive and out via the baggage hall. Then go left, up the escalator to the floor above, and security is at the far right of the ground floor. The ba flights depart from the a gates so once through security go down and through duty free and through passport control.

LUX is fairly small and efficient so you should be back at the gate within 5-10 minutes.