Gun ownership levels in the UK in the nineties weren't high enough for banning to have any significant impact on firearm crime.
The handgun ban was simply a knee-jerk reaction to Dunblane.
Legaly bought guns weren't a significant problem in the UK, because unlike the US, they were registered. You couldn't sell a gun without transfering ownership to another licenced owner, so there was always a track on where a particular gun went.
The rise in crime in the UK (and yes, the rises are worst in London) has more to do with the lower numbers of police, the constraints placed on the police, and the very poor clear-up rate for crime.
Even then, then murder rate in London is a third that of New York, and abouth one fifteenth that of Washington DC.