Yes, that is a shot of Earl Cameron in Pool Of London. I agree, there are some wonderful films listed.
"Barney SloaneThat's my new nameMy old one's a little more Italian."

Is your avatar a shot from "Pool of London"?
I think between all the contributors here we have assembled a powerful collection of titles. Some dross, but very little really.

Thanks London. I love British films of this period (and earlier), particularly directors like Robert Hamer, Basil Dearden, Seth Holt, Wendy Toye, Ken Hughes, Thorold Dickinson, Charles Crichton, Lewis Gilbert,Walter Forde etc.So much talent, and so many wonderful films. I 2nd your excellent choices of The Yellow Balloon, Woman In A Dressing Gown, Hell Drivers, and The Magnet, and would add
Hunted
(1952) and
The Long Memory
(1953). The period offered so many gems, the list is virtually endless.
"Barney SloaneThat's my new nameMy old one's a little more Italian."

Great list, CanterburyTale.

The following films (some perhaps already mentioned) depict the changing landscape and attitudes of Britain in the 1950's and 60's, particularly London.
Pool Of London -1951
A Place To Go-1964
No Trees In The Street -1959
The Secret Place -1957
The Wind Of Change -1961
Sapphire -1959
The Heart Within -1957
Flame In The Street -1961
Ten Bob In Winter -1963
Jemima And Johnny -1966
The Boys -1962
Nowhere To Go -1959
Victim-1961
There are, of course, many others,and this is not simply a list of favourite films, but these dramas specifically touch on subjects such as race and sexuality, as well as environmental changes, and are highly recommended.
"Barney SloaneThat's my new nameMy old one's a little more Italian."

Tom Jones, (1963)
Sons and Lovers, (1960)
Genevieve, (1953)
The Net, (1953)
The Green Glove, (1952)
Time Bomb, (aka Terror on a Train) (1953)
I thought that I could select a year, and chose 1957, for this:
https://.en.wikipedia.org/wiki/List_of_British_films_of_1957
If this works, perhaps you could type in other years. Good Luck!

Never Let Go (1960)

A Kid for Two Farthings

Robbery
The Servant
Blow-Up
Night And The City

Legend (2015)
http://www.imdb.com/board/13569230/?ref_=nm_flmg_act_9

A bit odd to call the 1960s a "post-war" era. Films made, or retroactively set, in the 1960s were typically in a different style and concerned with different issues than "post-war" movies.
But maybe that is your aim, to monitor the rapid changes that occurred around the change of decade?
Whatever, there are lots to enjoy from the recommendations already made.
I endorse the suggestion of "It Always Rains on Sunday" (1947). A detailed and accurate portrait of what is the true "post-war" era, in this case in the East End of London.
A modern film that captures the '40s well is "The Deep Blue Sea"(2011).
Others:
Dial M for Murder (1954) Alfred Hitchcock
Never Let Go (1960) John Guillermin
Repulsion (1965) Roman Polanski
Made in Dagenham (1999) authentically set in 1968
The Red Shoes (1948) - a masterpiece from Powell and Pressburger.
Peeping Tom (1960) by Michael Powell cinematically and thematically ushered in the '60s. Unfortunately, its being so ahead of its time effectively ended Powell's career.
Kes (1969) Ken Loach
Victim (1961)
Hell Drivers (1957)
The Yellow Balloon (1953)
Woman in a Dressing Gown (1957)
Accident (1967)
Robbery (1967)
The Magnet (1950)
Quartet (1948) and Trio (1950) are set in the 1940s with an array of British stars and starlets of the era, but are based on W.Somerset Maugham stories from the '30s.

Room at the Top (1959)
Look Back in Anger (1959)
The Entertainer (1960)
Saturday Night and Sunday Morning (1960)
A Taste of Honey (1961)
The Loneliness of the Long Distance Runner (1962)
A Kind of Loving (1962)
Billy Liar (1963)
This Sporting Life (1963)
"It is a good thing for an uneducated man to read books of quotations" Winston Churchill (1874-1965)

Night and the City /1950/ Jules Dasin
Man in The White Suit /1951/ Alexander Mackendrick
The Man Who Knew Too Much /1956/ Alfred Hitchcock
Village of The Dammed /1960/ Wolf Rilla
A Hard Days Night /1964/ Richard Lester
Help /1965/ Richard Lester
The Ipcress File /1965/ Sidnehy J. Furie
Two For The Road /1967/ Stanley Donen
If. /1968/ Lindsay Anderson
The Magic Christian /1969/ Joseph McGrath
The Prime of Miss Jean Brodie /1969/ Ronald Neame

these look perfect. many thanks!

Vera Drake (2004)
Ginger and Rosa (2012)
An Education (2009)

You're welcome. Enjoy.
If you're interested in films made immediately after the war, these might also work for you:
Hue and Cry (1947)
Passport to Pimlico (1949)
It Always Rains on Sunday (1947)

many thanks! excited to dig into these now.

The Green Man (1956)
An Inspector Calls (1954)
The Happiest Days of Your Life (1950)
The Man in the White Suit (1951)
Laughter in Paradise (1951)
The Ladykillers (1955)
The Lavender Hill Mob (1951)
School for Scoundrels (1960)
The Belles of St. Trinian's (1954)
I'm All Right Jack (1959)
Too Many Crooks (1959)