Hello Gideon,
if your kernel spec is hardwired to the old version, you'll have to replace the kernel spec with a new one. If your kernel spec just generally points at a "Julia environment" and uses whatever version happens to be installed there, it should work without changes. Unless it has a version number in the display name, then it would use the updated environment but still display the old version. That's assuming you updated Julia in place. If you installed the new version into a new "Julia environment", the old kernel spec doesn't know about that and has to be replaced.
It really depends on the specifics of your installation and kernel spec. As I'm not using Julia myself, I don't know what the default approach is there. Maybe you can figure it out by looking at the kernel spec. Or somebody else with more specific knowledge can help you further.
cheers and good luck,
Roland