Both Gently and Baccus are seen using a police radio in their cars, but neither of them display an antenna which, in those days, would have been clearly visible.
When Gently and Bacchus go looking for Hazel Joyce, the camera pans across to follow Bacchus from the car to the seafront. As it does so, a modern car (a Vauxhall Nova) is seen parked in a side road.
Before the UK adopted the Divorce Reform Act of 1969 (which instituted a no-fault divorce), to obtain a divorce required serious grounds such as adultery, cruelty, or incurable insanity. As such, the 1960s divorce rate was roughly 3 individuals per thousand married people. Not nearly as high as the show might lead you to believe.
In the WWII flashback scene towards the end, the then 17-year-old Frances is shown hooked up to a physiologic monitor. These were not commercially manufactured until the mid-1950s.