He initially worked for BBC Television in their Talks Department, where he was the Editor of the topical magazine programme Highlight and then co-devised and edited its more ambitious and better-remembered successor Tonight, which began in 1957.
He subsequently became involved in the establishment of the ITV northern franchise holder Yorkshire Television, becoming the company's first Director of Programmes and overseeing the creation of popular hits such as the soap opera Emmerdale Farm (from 1972).
Also in 1972, as ITV commissioned the broadcast of the first nationally televised darts tournament with the 1972 News of the World Darts Championship, Baverstock asked Sid Waddell to accompany presenter Peter Jones and commentator Dave Lanning at the 1972 News of the World Championship, to observe the action and to look into creating a programme based on indoor pub sports.
Former colleague Leonard Miall claimed in Baverstock's obituary in the Independent newspaper that the BBC Governors who interviewed him were "put off" by his "casual behaviour".
In 1957 Baverstock married Gillian Mary Waters, elder daughter of British children's author Enid Blyton, at St James's Church, Piccadilly.