Mock cockney certainly predates the current bunch of wannabees. The Sweeney TV series was guilty, particularly in popularising the over-pronounced T sound in phrases such as 'get it', which working class Londoners would pronounce 'ge i'.
Could Dick Van Dyke in Mary Poppins be an early edition? Or Audrey Hepburn's appalling mockney in My Fair Lady. (No disrespect, great film, but she sounds like she comes from Botswana) As to when it became "cool" (if that outdated phrase can be applied)...yep, probably the Sweeney. or Alf Garnett.
Cor blimey, I think allen and incitatus have got the wrong end of the stick. This section is phrase & sayings, I'm not talking about Mock cockney accents, I'm talking about mockney-which I have a feeling is more about getting cockney wrong. What do other people think? By the way I think Eliza Doolittle sounded nothing like a Botswanian.
I think the term was first coined about the time movies like Lock Stock.., Snatch, Love Honour and Obey, Face, Gangster No. 1 ad nausem were hitting the silver screen, often perpetrated by people like Guy Ritchie, clearly a middle class chap but one whom likes to perpetuate the image of being a real geezah. To wit - Guy Ritchie is a mockney: probably from Dartford or somewhere but wishes and acts like he is from Isle of Dogs.