His accent certainly doesn't sound overdone. Jamie Oliver sounds just like tens of thousands of people from Essex, and many people have even more extreme Essex accents than he does.
Whether he chooses to exaggerate his accent in order to reinforce his image - that of a practical, down-to-earth ordinary bloke - who knows? The world of celebrity, like most things nowadays, is all about branding. As a brand, he is marketed as a counterpoint to the stereotypical chef, full of pretensions and fancy French phrases. By contrast, Jamie is 'a regular guy' - hence the cheery manner, bish-bash-bosh style of cooking, and working-class accent. It's all part of the package.
If his manager told him to up the Essex-boy sound a notch or two, I don't blame him, or Jamie. Nor is he the first to have been accused of this - Madonna's film director ex-husband Guy Ritchie and classical violinist Nigel Kennedy have both been criticised for their rather-too-deliberate working-class accents. We all make choices about the public image that we present to the world, and, considering how much deceit and duplicity there is in the world, I consider exaggerating your Essex phonemes to be a fairly minor crime.