This has always been something that's the subject of much debate and dispute. When I was growing up (in Bow), all the local 'Cockneys' insisted that the 'Bow Bells' were those of Bow Church, and not, as popular legend has it, those of the bells of St Mary Le Bow in Cheapside. Either way, it appears I'm a 'Cockney*' anyway:
"The region in which cockneys are thought to reside is not clearly defined. Originally, when London consisted of little more than the City, the term applied to all Londoners, but as the city grew this was replaced by less universal definitions. A common view is that in order to be a cockney, one must have been born within earshot of Bow Bells, the bells of St Mary-le-Bow,[23]which were cast at the Whitechapel Bell Foundry. However, the church of St Mary-le-Bow was destroyed in 1666 by the Great Fire of London and rebuilt by Sir Christopher Wren. Although the bells were destroyed again in 1941 in the Blitz, they had fallen silent on 13 June 1940 as part of the British anti-invasion preparations of World War II. Before they were replaced in 1961, there was a period when, by the "within earshot" definition, no "Bow Bell" cockneys could be born.[24] The use of such a literal definition produces other problems, since the area around the church is no longer residential and the noise of the area makes it unlikely that many people would now be born within earshot of the bells,[25]
The terms “East End of London” and “within the sound of bow bells” are used interchangeably, and the bells are a symbol of East End identity. However the Bow Bells definition reflects the earlier definition of Cockney as relating to all Londoners (at a time when London barely extended beyond the square mile. The use of the term to describe all Londoners generally, however, survived into the 19th century[10] before becoming restricted to the working class and their particular accent. The term is now used loosely to describe all East Londoners, irrespective of their speech.
A study was carried out by the City in 2000 to see how far away Bow Bells could be heard,[26] and it was estimated that the bells would have been heard up to six miles to the east, five miles to the north, three miles to the south, and four miles to the west. According to the legend of Dick Whittington the bells could once be heard from as far away as the Highgate Archway (4.5 miles north). Based on a definition of the bells audible range, all East Enders are cockneys, but not all cockneys are East Enders; though whereas an East Ender would be likely to proudly claim that entitlement, a resident of west, north or south London would be less likely to. The traditional core districts of the East End are Bethnal Green,Whitechapel, Spitalfields, Stepney, Wapping, Limehouse, Poplar, Haggerston, Aldgate, Shoreditch, Millwall, Cubitt Town, Hackney, Hoxton, Bow and Mile End. The area north of the Thames gradually expanded to include East Ham,Stratford, Leyton, West Ham and Plaistow as more land was built upon." -From Wiki.
*I don't, personally, identify as a 'Cockney', although many people have considered me such throughout my life. I am, however, definitely a Londoner. Many tribes have settled and lived here, and many more will come. And London will welcome them all, as it has done for centuries, millenia really. The Romans, the Danes, the Anglo Saxons, the Normans, the Jews at various times, the Dutch, the French, the Huguenots, Italians, Bangladeshis, Indians, West Africans, East Africans, West Indians, Turks, Poles, Romanians, Colombians, Brazilians, Afghans, Syrians, Swedes, Norwegians, South Africans, Australians, even people from The North™; all and many, many more have been and left their mark. The more the merrier!