I found this short history of the BBC Pebble Mill site on a Facebook Page, ‘Birmingham Back in The Day’. Here is the link to it: https://www.facebook.com/groups/1177026819016145/permalink/1645644118821077/?pnref=story.
“Pebble Mill was a fulling mill in the 16th century. Standing on the Bournbrook, it was also known as Kings Mill or Kynges Mill after John Kynge. On his death in 1557 this mill together with Over Mill at Edgbaston passed to his son, Roger Kynge. The mill was used for blade grinding in the mid-17th century and was a corn mill by the mid-19th century.
The millpool, which was drained in 1883 allegedly after a high incidence of suicides, became the site of the BBC Pebble Mill studios from c1960. The building was demolished in 2005 when services transferred to the Mailbox in the City Centre.”