The overall rate of homicide in the UK is about 1.1 per hundred thousand population per year recently. It peaked at 2.1/100K in 2002. The U. S. is at about 4.7/100K. It was about at low as the UK in 1900, and has been as high as 10.2/100K in 1980, with secondary and tertiary peaks at 9.8/100k in 1991, and 9.5/100K in 1993. It has been trending down since then. Switzerland, with lots of privately controlled full auto selector switch true assault rifles, is at about 0.6 per hundred thousand population per year.
There just isn't any direct correlation between private gun ownership and violent death rates throughout the world. This page on the Wikipedia
is useful to look at. When you pick a year and sort the columns by homicide rate, you may be flabbergasted by how many countries are higher and sometimes much
higher in their homicide rates than the U.S. is. Let's just say that South America is not the safest place to be.
I believe what has increased in the UK is the portion of gun crimes, and that major cities have much higher crime rates than the country as a whole. The same is true here, as anyone living in Chicago knows. One has to be careful with what specifics narrow the case, and to compare apples to apples.