I managed to get in a gun control debate today. One of the comments made was about how the gun laws in the UK affected their violent crime rates... and while I wasn't able to push back on that as well as I'd hoped in the moment, there are a lot of interesting details I found when I was able to look up the statistics later.
So. The short answer to the question "did violent crime in the UK go up after they banned handguns?" is "no, probably not".
Of course, there are a lot of details we can dig into. The first - and the one that any gun rights advocates are probably thinking of as they read that - is that the number of offenses reported to the police has gone up on several different scales - in the last five years, in the last 20, and overall since the start of the data set I cited above.
That doesn't seem to imply that crime is going down. However, there are other indicators. There are also crime victimization surveys and other types of scientific study... and those indicate that violent crime has gone down quite a bit in the last 20 years.
So which is right? Neither, really - both data sets have their limitations. The victimization surveys and studies are almost certainly more accurate for the types of crime and population groups for which they're able to get enough data, since they can evade some reasons why people might not want to or be able to report crimes in order to get a more comprehensive picture. Unfortunately, their data is likely less accurate for less frequent crimes and is mostly useless for some groups of the population. On the other hand, the number of crimes reported to the police makes for a very accurate minimum baseline, but changes in what types of crimes are counted, under-reporting of certain crimes, and the possibility that reporting rates might change without the overall level of crime changing make it tricky to even be certain the trend in reports accurately reflects trends in the overall crime rate, much less to assign causation for those trends.
... Of course, just because neither is fully correct doesn't mean that we can't make a judgment call as to which more accurately reflects reality. And when there have been numerous changes to reporting criteria, surveys indicating higher reporting rates, and other data (such as hospital admissions) suggesting less crime, I think the overall violent crime rate in the UK has gone down since the mid-90s - hence my short answer above.