Insofar as some parts of it (so called "City's Cash") function as a private entity from monies "built up over the centuries" to such extent as to make it exempt from the Freedom of Information Act, then who controls these private components?
City’s Cash is a fund of the City of London Corporation
City’s Cash is managed by committees of the City Corporation, with Membership of the committees drawn from the 25 Members of the Court of Aldermen and the 100 Members of the Court of Common Council. Members of the Court of Aldermen and Court of Common Council are elected by registered voters (both residents and businesses) within the City of London. In determining appointments to committees, the Court of Aldermen and Court of Common Council will take into consideration, amongst other things, any particular expertise and knowledge of the Aldermen and Members.