The most well known South London firms were/are the Arif, Brindle and the Daley family. The Arifs are Turkish Cypriots, the Brindles and the Daleys are both part-Irish (Brindles on their mother's side, Daleys on their father's side). Most, if not all, white working class families in East and South London will have some Irish blood, but few are fully Irish. The Adams of North London are an exception (Islington has long been a settlement for more recent Irish immigrants), but families in the east and south will commonly have one Irish parent or at least an Irish grandparent.
It's true, a lot of the London-based crime firms are in business with the Real IRA. Even the Canning Town families such as the Hunts, who as far as I know are not of Irish descent, frequently did business with some criminal minded cells in the IRA concerning the drug and weapon trade.