I'm the lead author of the study, and I would be happy to discuss the findings here with anyone who is interested. A pdf of the paper can be downloaded from my website:
http://www.cpbell.co.uk/home/pubs.
Just to respond to a couple of points already posted. John Cantelo asks why Sparrows were not depleted by Sparrowhawks when they were common in the 1940s. As far as I can tell, Sparrowhawks were never common in cities prior to the late 1980s. Remember that cities were very polluted prior to the Clean Air Acts, and Sparrowhawks would also have been shy of humans since they were persecuted in the countryside. In the country, Sparrows would formerly have been used to Sparrowhawk attacks. However, with Sparrowhawks absent for several decades after the 1950s Sparrows may have lost their predator averse behaviour, making them easy pickings when the predator returned.
Several posters have mentioned that Sparrows remain common in their neighbourhood despite hunting Sparrowhawks. We found this was often the case when analysing the data we used for the study - the BTO's Garden Bird Feeding Survey. However, when averaged over all sites the effect of Sparrowhawk presence is quite striking. This may not show up in individual sites, each of which has its own special characteristics which affect the way in which the predator and prey interact.