In the UK we have the capability[1][2] to recycle all the traditional PE lined coffee cups we use here.
There are also quite a few collection services[3][4]. One of the challenges is separating the waste ready for recycling so having collection points in cafes, offices, and railway stations[5] is a really good start.
There are also some new cups with reduce the amount of plastic in a cup so that it's below the 5% contamination level fr general paper recycling. Personally I think this is not a solution as it's just gaming the system buy encouraging contamination of the waste stream.