The decline of House Sparrows seems to be rather complex. Studies by the British Trust for Ornithology over a 30 year period suggest the following are probably the main ones:
Reduction of brownfield sites in urban areas and invertebrates in general in all areas.
Loss of nest sites modern houses/buildings do not have the gaps and crevices which are good for nest sites.
Pollution in urban areas.
Predation particularly by Sparrowhawks and Magpies doesn't seem to be important. Being hole nesters Magpies taking young is very limited and strangely Sparrows were found in good numbers where Sparrowhawks were also found. No definitive link to cats was found/proven.