Jimmy Savile (in-universe)

Jimmy Savile was a DJ.

Savile hosted the 13 May 1965 instalment of Top of the Pops, in which The Beatles performed their song "Ticket to Ride". The First Doctor and his companions Vicki Pallister, Ian Chesterton and Barbara Wright watched it via the Time-Space Visualiser. (TV: The Chase)

Kitty told Ben Jackson that the First Doctor looked like "that disc jockey". (TV: The War Machines)

On 30 June 1975, Jason listened to Jimmy Savile on the radio. (PROSE: Cuckoo-Spit)

On Red Nose Day 1991, Tom approached Roger Nellie with the idea of introducing the new Comic Relief book, with Mellie replying that he was happy to do something for the kids, and that it had never done Jimmy Savile any harm. (COMIC: )

Behind the scenes
Jimmy Savile was a broadcaster employed by the BBC for decades. In A Fix with Sontarans, a Doctor Who crossover produced as part of a 1985 episode of his show Jim'll Fix It, Savile portrayed himself. His scene was cut in the re-edited 2022 video release because the real Savile's prolific sex crimes had since become public knowledge.

Although it would not have been widely known at the time, it can be assumed that Roger Mellie noting that doing something "for the kids" had never done Saville any harm was a reference to this.

Information from invalid sources
After the Doctor defended the TARDIS from two Sontarans with the help of Tegan Jovanka and a young boy named Gareth Jenkins, the scanner displayed an image of Savile, which Tegan called "monstrous", and the Doctor said was "revolting". The TARDIS doors swung abruptly open and Savile entered. He shook the Doctor's hand, kissed Tegan's (at which the Doctor frowned), and greeted Gareth, recognising him as "a small clone-type Doctor". Savile asked the Doctor, "Did he pass the test?" and the Doctor replied, "With flying colours." Savile complimented Gareth's nan for her work on his outfit, then handed the Doctor a medal to give Gareth, asking if he could "place this over his bonce without disappearing his bonce with your amazing powers". The Doctor complied, then handed Savile the dead Sontarans' meson gun to give to Gareth, which Savile called "a nice gesture". (TV: A Fix with Sontarans)