I agree, it does sound a bit false and affected, but I think that's probably right for the times. If you listen to old radio programmes/watch old films or TV programmes up to the early 60s, almost everybody is using that sort of accent, unless they're a comedy character like Gracie Fields or Norman Wisdom or the Carry On Team. The odd film during the war allows characters to have regional accents to make a point that the whole country was pulling together in the war effort, but even then they're usually chirpy Cockneys or reliable NCO or foreman types, not officers or managers or professional people.
If CTM aims to be realistic, this series and the next one will have to reflect that there is a huge social change coming to Britain in the mid 60s. We've had mention of the Rolling Stones but The Beatles should be omnipresent in this series, and from that point on regional accents suddenly became supercool. That, and the emergence of the first generation of working class children who'd got a decent secondary education and proceeded to university or art college or got jobs with large employers who would train them up, send them off to college on day release and promote them on merit. Social mobility was about to be the highest it had ever been in the UK and to our great shame it's now heading back to where it was.
If it went on to the end of the 60s, we'd see huge legal changes - divorce law reformed, male homosexuality no longer criminalised, abortion legal in some circumstances, Equal Pay Act, Sex Discrimination Act, Race Relations Act. We should also be about to see a slow decline in the number of unplanned pregnancies and unmarried girls pressured to give their babies up for adoption or get married. We'd probably also see a rise in sexually transmitted diseases and drug problems.
What we should be seeing more of is families fractured by well-intentioned but poorly executed rehousing schemes. Lots of new housing was in new towns or outer London suburbs and young couples with new babies often ended up a long, long way from the family support that had always been a big feature of East End life.