Flip Jackson did a first aid course with St John's Ambulance. (AUDIO: Static)
The Sixth Doctor asked Peri Brown if she had done St John's Ambulance. She told him that she had not. (AUDIO: Year of the Pig)
The Doctor's TARDIS sometimes had a St John's Ambulance sticker on its door. (TV: An Unearthly Child, The Eleventh Hour, etc.) However, after regenerating on Desolation, the TARDIS did away with the sticker. (TV: The Ghost Monument [+]Chris Chibnall, Doctor Who series 11 (BBC One, 2018).)