The cardboard cling film core was a bust - as I had anticipated. It just frayed horribly when I took a hacksaw to it. It didn't cost me to try, though. So I've gone back to the method I used for the original iteration of the steering wheel, albeit with refinements.
Started out with a change of card type - used the same stuff that's gone into the structural elements of the model, so I know it's not going to flex. Plus I went back to using superglue - that roket glue bonds well, but doesn't dry rigid. I worked outside to stick the bits together in order to minimise my exposure, and it's now in the hallway drying out in preparation for the next stage.
Another plus is this new version isn't dished, which does make the construction a lot simpler. And I used some photos of a Mk2 Cortina steering wheel as a reference this time. One of the mistakes I made last time was assuming that the spokes were equidistant. But so far, things already look much more promising.