This can be a bit of a minefield.
When I used to do my folk radio show between 1983-1993 on Mercia FM in Coventry, the radio station used to sponsor Warwick Folk Festival. As I was also doing the sound for the festival, I used to record all the main stage performances (I still have all those tapes - must go through them some time!), and much of this material was used in a couple of 'festival specials' each year. We even did a live broadcast one year of Show of Hands.

Normally, permission would need to be sought from each artist, and an appropriate fee paid, but we got round this by the festival including a clause in the artists' contracts along the lines of "your performance may be recorded for broadcast on the local radio station and the agreed fee includes the rights to broadcast". Obviously, the station paid the normal PRS royalties.

We never had any problem - it's OK if you are upfront about it.

We also did Cropredy one year, where we sponsored the festival in return for being able to use the recordings (still got those as well!!).

We've even released some of this stuff (in collaboration with the artists, of course). Most notable being Gordon Giltrap's "On a Summer's Night", which became his biggest selling album at the time.

Mercia was/is a commercial station - you wouldn't get them doing something like that these days!

The BBC tend to need to do everything by the book, whereas we were able to come up with an informal arrangement that worked to everybody's benefit.