I wasn't sure how much brand naming I should do!
It's called Araldite Multi-Pupose Putty, and at least Loctite do a near identical product (though that might not work). It is supposed to be heat resistant to 120° C . It dries to something very similar to the plastic the plate mountings are made of too.
A big blob of it held a cracked copper hot water pipe for a couple of years (and it was still rock solid when the pipe was taken out), so I thought it was worth a try (nothing to lose). The straightners have been in daily use since and have also been "bounced" about 18 inches to the carpet a couple of times (accidentally) and they are still holding perfectly.