After some research on this I've found some TV shows that ran concurrently with season one.

We'll start off with Ready Steady Go which was a contemporary music show, ITV's precursor to Top of the Pops if you like. With its slogan "The weekend starts here" RSG was presented by Keith Fordyce and Cathy McGowan, herself a teenager plucked from the streets to present the show. The show ran for three years from 1963 to 66. The edition of 22nd November '63 featured Freddie and the Dreamers, Gerry and the Pacemakers and The Rolling Stones.

That Was The Week That Was, was a satirical show lampooning the politicians of the day, TW 3 made household names of David Frost and Millicent MArtin, who sang the theme tune.
For the edition broadcast on Saturday 23 November 1963, the day after the assassination of United States President John F. Kennedy, TW3 produced a shortened 20-minute programme with no satire, reflecting on the loss, including a contribution from Dame Sybil Thorndike and the tribute song "In the Summer of His Years" sung by Martin with lyrics by Herbert Kretzmer. This edition was screened on NBC in the US the following day, and the soundtrack was released by Decca Records. BBC presenter Richard Dimbleby, who broadcast the president's funeral from Washington, said that the regular programme was scrapped when news of the assassination was received and that the programme was a good expression of the sorrow felt in Britain

The Avengers, well I think we all know this one. The Avengers initially focused on Dr. David Keel (Ian Hendry) and his assistant John Steed (Patrick Macnee). And was a spin off from Hendry's own series Police Surgeon, Hendry left after the first series and Steed became the main character, partnered with a succession of assistants. Steed's most famous assistants were intelligent, stylish and assertive women usually clad in leather: Cathy Gale (Honor Blackman), Emma Peel (Diana Rigg), and later Tara King (Linda Thorson). Later episodes increasingly incorporated elements of science fiction and fantasy, parody and British eccentricity. The Avengers ran from 1961 until 1969, screening as one hour episodes its entire run.
The episode that aired on 23rd November 1963 was The Medicine Men in which Steed and Cathy Gale discover that somebody is flooding the market with cheap imitations of medical products being manufactured by the Willis-Sopwith Pharmaceutical Company. Furthermore, there is a plan to distribute poisoned medicines in an oil-rich Middle Eastern country to start a revolution and drive out the British companies there. This episode was written by Malcolm Hulke and also featured Peter Barkworth.

Other shows that aired around this time were the obligatory ITC shows like The Saint, still in its black and white era, Dixon of Dock Green, Doctor Finlay's Casebook and Sunday Night At The London Palladium.
American imports included the top sixties series The Fugitive starring David Janssen as Richard Kimble, a doctor wrongly accused of the murder of his wife, he escapes custody and must stay ahead of the police to find the real killer, a "one armed man" who he saw commit the killing. This was a pot boiler of a series which kept sixties audiences enthralled until 1967 when the real killer was revealed.

I invite your comments on contemporary TV, and I'll do some more research on this and see if I can find some more.