Edit: The solution in my case was to run these commands:

sudo apt update
sudo apt install linux-image-rpi-v8 linux-headers-rpi-v8

The new kernel wasn't used after rebooting though, so I had to edit /boot/config.txt and add:

# New kernel (test)
kernel=vmlinuz-6.12.75+rpt-rpi-v8
initramfs initrd.img-6.12.75+rpt-rpi-v8 followkernel

I'll figure out how to make it so I don't have to manually keep config.txt updated but for now, I'm happy to have a process to stay up to date with current kernels.

/edit

Reading about the latest Copy Fail security vulnerability I realised I was overdue a check on my system. It turns out I had even bigger problems - I was still on Debian 10/Buster.

To solve it I updated my apt list to contain: deb http://archive.raspberrypi.org/debian/ bookworm main

But after doing this, updating and doing a full upgrade, my kernel is still 6.1.21.

I'm now reading that I should not have upgraded from bullseye to bookworm, so what is the easiest way forward from here? If there is a documented set of steps for this scenario I'd appreciate a link. I have a lot of services and scripts so reinstalling from scratch is not really an option.

uname -a
Linux 6.1.21-v8+ #1642 SMP PREEMPT Mon Apr 3 17:24:16 BST 2023 aarch64 GNU/Linux.

cat /etc/os-release
PRETTY_NAME="Debian GNU/Linux 12 (bookworm)"``___