Ah. Does it in any of the three situations mentioned does it? very unlikely to be just one bulb then. The typical scenario for this is that one of the main flasher bulbs - so either a front or rear on one side - will be blown or maybe a bulb of the wrong wattage. This affects the current draw in that circuit which forces the flasher unit to "blink" faster. But this would be immediately obvious in that one of the flashers on that side wouldn't be flashing. It's just possible that bulbs of the wrong wattage may have been fitted - normally 21 watts - so you need to take all bulbs out and check the wattage, which is usually stamped into the metal of the bayonet filling just under the glass of the bulb. Has someone tried to do an LED bulb conversion? there's lots about troubles if you try to do this.
I don't know the 500 as well as I know the older Pandas but I believe the flasher unit is an integral part of the unit and is soldered directly into the printed circuit board. If you need to replace this I'd be very wary about buying a "cheap" option. Again, I don't know about your particular car, but older flasher units were current dependent for their speed of flashing so bulb problems caused variable flashing rates. Quite some years ago some manufacturers started producing flasher units which were electronically controlled for speed of flashing and this type was not dependent on current draw - as far as I know - so, with this type, the bulbs didn't affect flashing speed. With yours being an integral part of the circuit board, ie not a plug in, I wouldn't be surprised to find it's one of these.
As it's by far the easiest thing to do, I'd start by checking out all the flasher bulbs for correct wattage. Incidentally, the wee repeaters on the wings are quite low current consumers so usually have minimal impact on this sort of problem. In other words are unlikely to be a problem re passing an MOT?
Providing the outside bulbs are flashing at the right rate (I seem to remember between one and two flashes per second?) I don't think the dashboard repeater comes into it - unless it's triggering the check engine light to illuminate?