As a non British person I would think the center of the city would be some point equidistant centrally between the boundaries of the original roman city wall, in the square mile, but the actual center of the city seems to have migrated since then.
That would be the centre of the City, but not necessarily of the city.
E.g. Waterloo Station would be considered South London but is actually at the same latitude as Buckingham Palace!
Hence why a Londoner would never describe Vauxhall as "Central".
Londoners would generally discount any part of this map south of the river from "Central London":
https://en.wikipedia.org/wiki/Central_London#/media/File:Ope...