What do you mean by the City? Strictly, that refers to the so-called "square mile" of the medieval city, which is administered by the City of London Corporation which hasn't changed much in hundreds of years. Briefly, there was and is the Common Council made up of Common Councilmen, Aldermen and the Lord Mayor.

Outside the City proper, local government at the start of the 19th century was, like everywhere in England, almost entirely organised by Parishes. At this stage the Vestries- parish councils- had both ecclesiastical and secular responsiblities. Gradually parishes lost powers and responsibilities- they no longer appointed constables once the Metropolitan Police was established in 1829, and support of the poor was transferred to the new poor law unions in 1837.

As London grew, it became evident that such things as sewage systems and highways needed to be organised London-wide rather than by parish. In 1855 the Metropolitan Board of Works was set up covering a large area of central London, and many of the Vestries' powers were transferred to it. Also, the vestries' civil functions were seperated from the ecclesiastical ones.

The situation remained like that until 1889 when the London County Council was formed, replacing the MBW with a properly elected body. The vestries remained for a few years until they were grouped into Metropolitan Boroughs at the start of the 20th century.

This book looks like it covers the ground you need.