I use a mug because the 50c off pays off within a month and the mug keeps the coffee warm longer than the paper cup and no, the barista pours my coffee directly into the mug
Also, some places do brew or mix in a paper cup first, some will brew directly into your cup.
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.
[1] https://www.dssmith.com/recycling/insights/case-studies/case...
[2] https://www.recyclingtoday.com/article/james-cropper-paper-m...
[3] https://thefirstmile.co.uk/service/coffee-cup-recycling
[4] https://www.veolia.co.uk/services/coffee-cup-recycling
[5] https://www.networkrail.co.uk/news/network-rail-introduces-c...