That's true. We do have the Firearms Directive (EU) 2021/555 - which pretty much does what the British law did... but it did cause some controversy. The Swiss gave themselves an exemption because they have a gun culture, then the Czech's objected to the Swiss giving themselves some freedoms not guaranteed to everyone else in the EU. The Swiss then held a referendum and collectively voted for tighter restrictions on guns in line with the directive.

The EU might be divided on a lot of things, but gun control doesn't seem to be one of them. It's a peculiarly American problem. :o)