I wish you'd said you had a '94 car earlier, people never put what car they are talking about. Mini changed a lot over 41 years and the late indicator system is different from the vast majority.
The line fuse runs all the flashing lights, the fuse box fuse only runs a relay in this system (not a flasher unit).
When I said...
you didn't check the one that you know isn't working then? Have you checked the failed lamp at all? If the holder is broken, wet or dirty that will probably be causing your problem. Other than that it could be the flasher unit as you suspect but electronic ones like yours tend not to fail or it could be the hazard switch.Are any of the bulbs blown? They can blow dead short.
<Don't know why we bother, people don't seem to bother reading the replies>