Spicy hot works, and I'm sure I've heard it before, though not sure whether in the UK or US. "Chilli* hot" would be the alternative that comes to mind as a runner up, as it tags both the source and sensation. And, as an aside, yes, cinnamon can be spicy hot. There is a whole genre of candy that appears this time of year to celebrate that fact- cinnamon hearts.
*or whatever your preferred spelling is to refer to products of the capsicum plant... chili, chile, chilli, etc...