Thanks to @humdinger, the very comprehensive ZX Spectrum emulator FUSE is available in the Haiku Ports repository.
You can install from the command line with:
pkgman install fuse
Alternatively, install it from HaikuDepot.
Run from a terminal with:
Read the docs on the fuse website FUSE website for detailed usage, but to get started you just need to know that F1 opens the main menu. From here you can open tape and snapshot files of various formats (get them from the World of Spectrum) and things like setting the graphics filter to give it an old TV effect and increase the window size. Joysticks are working, just select the type in ‘peripherals’, e.g. Kempston.
Now, back to Jet Set Willy!