The typical American marquee is a large, three-sided sign extending to the street in front of the entrance to a cinema, theater, or other entertainment venue. That the sign lights the entrance like an airport runway or shelters theatergoers from the weather is merely incidental:
Movable letters may provide information about current performances or there may be custom graphics for each major production.
After scouring Google images for West End theatres, the closest I've found to having an American-style marquee is the Palace:
Here, however, as much as the elaborate signage visually overpowers the arcade extending across the entire façade of the building, they are architecturally distinct structures.
Other theatres have large signs extending streetward from an upper story or flat across the upper façade, but nothing resembling the American sign cum awning one sees on Broadway.
The reason the British don't have a word for marquee is simply because they don't need one. Otherwise, if you say marquee to someone from the UK, this is what they'll have in mind: