Update: Those intelligent people in the Intelligence team at the Greater London Authority have now made a better map of population change between 2001 and 2011, which I think you should look at rather than mine (there's more GLA analysis of the Census here).
The GLA map is better because (a) it uses ward boundaries, which unlike the statistical boundaries I used have not changed over time and therefore offer a like-for-like comparison; (b) it compares Census 2011 population to the 2001 mid-year population, which the GLA thinks is a more reliable figure than the 2001 Census figure, and (c) it's interactive! So I've put my map below the fold here, just for reference.
The data is all from ONS, analysed in Excel and mapped in QGIS. In a few cases the MSOAs in 2011 don't match up with those in 2001, so I had to split or merge the old areas to match the new ones. That means that in a very small (less than 1%) of cases the change I calculated isn't completely accurate. Hopefully ONS, who have all the raw, address-level data, will do a complete analysis of change at some point.
The largest percentage changes in population were in former docks areas where there wasn't much population in 2001 but there was space to build new housing. At first glance the largest falls in population seem to have been in areas where social housing estates are being regenerated and people have been moved out while the work is being done, though this needs a bit more checking. There were also falls in several parts of Kensington and Chelsea, the most expensive borough in London. Overall I think the key point here is that new housing supply is going in the cheaper places where there is unused space and not many people to object, but not where people really want to live as indicated by prices.