OS OpenData Locator has wrong name

In OpenStreetMap you can use any tags you like -- there's no such thing as an "invalid" tag. Your only responsibilities are to make sure your mapping and tags accurately reflect the real world, and are verifiable by other mappers.

As such it's perfectly fine to add not:name=* tags to roads that OS has the wrong name for. However, it's also fine not to bother -- the ITO OSM analysis tool that produces the layer you're looking at is just that -- a tool, and not an absolute arbiter of what's right and wrong. If you're happy that you've named roads in your area accurately, you can ignore any quality tool like this. They're only meant to be a guide, not instructions.

I agree in principal that you are free to ignore the ITO OSM analysis tool and not bother adding not:name=* tags - but if you have gone to the trouble of checking the names on the ground and found that they do not match OS Locator names then please do add a suitable not:name tag.

This will save other OSMers repeating your hard work and gives a clearer overview of the map quality. The not:name tags are also fed back to the OS, which helps raise the profile of OSM with them and demonstrates that the relationship is mutually beneficial.