I liked peakbagger.com's chart of the change in the rank order of US city population's that I shamelessly copied it below for London borough populations since 1801 (click for a bigger version). There are 33 boroughs (including the City of London), and in 1801 the largest population is in Westminster and the lowest in Brent, while by 2001 Croydon was largest and the City smallest.
Anyway, it's tempting to file this under 'pretty but not very useful', but I think you can still pick up three notable periods of change.
First, in the late 19th century the suburbs start to grow, presumably mainly as a result of the expansion of the railways.
There is relative stasis in the 1910s and 20s, but from the 1940s to the 1970s there is huge change, again reflecting suburban growth driven by changes in in transport, this time the advent of mass car ownership and huge investment in the road network.
Finally, the last two decades have seen something of a reversal, with inner London boroughs like Hackney, Southwark and Lambeth returning to growth and moving back up the ranks. I'd say this latter trend has continued in the years since 2001: most new housing growth has been intensive, i.e. densification and re-use of former industrial sites in the inner areas, rather than extensive, i.e. expanding the boundaries of London into previously undeveloped areas.
There are many striking trends for individual boroughs, but perhaps the most remarkable is Tower Hamlets, which was the first or second most populous borough until 1939, then lost two thirds of its population to rank 30th by 1971, before bouncing back to 23rd by 2001. Since then Tower Hamlets has probably seen more new homes built than any other borough, so expect it to have risen further up the ladder come 2011.