Radiohead sometimes use a click, but only judiciously.
They almost never do in live performance. In Radiohead’s early days (Pablo Honey, The Bends), they really considered themselves to be primarily a live band, and that has carried through to this day in how they perform (if not in how they make records). Phil has an extremely good sense of time, so as long as he’s playing then there’s little need for a click.
The only case of click use that I’m aware of is Little By Little on the 2012 tour. The band had a lot of trouble playing that song, and haven’t played it since part-way through the 2012 tour. The studio recording relied really heavily on complex layers of loops, and Radiohead had only recently started playing with a second drummer, so it’s no surprise that they had so much difficulty adapting it for live performance.
In the studio, Radiohead have occasionally used clicks. Their songs would suffer if clicks were used for the whole track, but there are cases (such as establishing the tempo when re-recording a chorus or verse) where a click can make the process a lot easier. I can’t find the source now, but Nigel once mentioned that he would sometimes use a click for the first couple of measures to establish the starting tempo, then turn it off.