Fetlar, cheers and yes, you have hit the nail on the head with this one as Southwark I've seen listed as Greenwich, St Olave, St Saviour and a couple of others. Meaning he could be registered with any of them.
I put a request in last night to Births and deaths for a birth cert on an open search, namely his details, parents, approx year which they search one year either side of and birth area London - not sure if it will work. If it doesn't then it will be the case of picking your aforementioned list off one at a time, in order of most likely.
Andy