I can't speak to how things work in the UK. However, I imagine it's at least similar to the U.S.
In the U.S., most publishing for popular music is administered through agencies like ASCAP and BMI. Those agencies grant blanket licenses to radio stations and stores and other places where music is publicly performed. The blanket license makes things much simpler. Otherwise someone at a grocery store would have to keep track of what songs get played at every store every day and then figure out how much to pay for each performance. It would be a record keeping nightmare. The licenses are issued based on the revenue of the licensee (on average, a bit less than 2% of revenue). The licensee still keeps logs of what is played, and ASCAP and BMI will do spot checks. However, at the end ot the day, the revenue from blanket licenses is distributed by how much airplay each song gets.
At the end of the day, the amount paid per play is in the 6-12 cent range. If you've got a song that gets played 10 million x per year and you're getting 10 cents per play, then you're getting $1 million per year. Reasonable. If that same songwriter is getting 30 GBP per play, he'd be earning 300 million GBP per year, which is obviously absurd.