Jeavons portrayed Uriah Heep in the BBC's "David Copperfield" (1966). He appeared in a host of 1960s and 1970s TV programmes including "Doctor Who" (in "The Underwater Menace"), "Adam Adamant Lives!" as a murderous fashion designer, as the undertaker Shadrack in "Billy Liar" (1973), as businessman Leonard Gold in "The Sweeney" (in the 1978 episode "The Bigger They Are"), as shop owner Ellery in "Shoestring" in the episode "Where was I?" (1980) and "The Avengers" (in "A Touch of Brimstone" and "The Winged Avenger"). Pete Stampede and Alan Hayes wrote of Jeavons in the latter series as "one of those under-rated, ever-present supporting actors who "never" turn in a bad performance." On children's TV, he hosted "Play School" for a time, and read "The Black Vicar" on "Jackanory". He also appeared in the 1981 Doctor Who spin-off "K-9 and Company", and he narrated two BBC children's animated series, namely "Barnaby" and "Joe".