“Lift” is such a sensible way to say “elevator,” if only because it has three fewer syllables, but I have never encountered it in the U.S., maybe because it is such a quintessential Britishism.
Never encountered it until now, that is. Here’s what I saw on the lower level of the Chelsea Market in New York City:
Was “lift” used here merely because “elevator” wouldn’t fit? Or is this the harbinger of a “lift” incursion on U.S. shores? Only time will tell.
notoneoffbritishisms.com 
Concourse fits so elevator will too, OK it will be a lot smaller. It would be interesting to find out why Lift was chosen. I love the arrow underscore broken for the ‘f’.
Not entirely sure what it means by “concourse”. Dictionary definition is:
1 A large open area inside or in front of a public building. ‘a station concourse’
2 formal, A crowd or assembly of people. ‘a vast concourse of onlookers’
I remember Chelsea market as being a crowded area of shops with no large open areas (but I did love it). The second definition is not relevant in the context of lift signage.
In a large railway station, It’s the open area inside before you get to the platform barriers. I’ve never heard of it in connection with a market.
Right, I’m not sure if “concourse” is the right word. The main floor of the market (to which the sign refers) is basically a long and fairly spacious corridor running from the 10th Avenue entrance to the 9th avenue entrance, with shops, food stands and restaurants on either side.
Does the US use lift as in ski lift and fork lift?
Yes.