I completely erased the card and reinstalled everything from scratch so everything's tickety-boo.
Yes, it does have inittab after all. After a bit of mucking about I made this change to it and it worked, so thank you very much. I've created an image of the installation with everything working, just in case of disaster.
Now I just have to install Ant's code and (maybe) it's done!
"We all have flaws, and mine is being wicked." James Thurber, The Thirteen Clocks 1951
You *should* now be able to just add the script you want to run to /etc/profile and it should run. That is if you're logging in automatically to a virtual terminal and not using a window manager. Ant's talk of window managers has me confused.
Edit: Although Ant's approach of running it as a daemon will make sure it comes back up if it crashes, which adding it to /etc/profile won't. So it's a better solution.