Here in the colonies where Mini stockists are few and far in between, I tend
to keep known good spares laying around.

So my point is if you can try another known good flasher unit.

Good luck as you get frustrated just keep in mind this is going to be a
simple fix as opposed to the same thing in a, gasp, 2006 MINI.

I would 'hazard' a guess (pun intended :-) ) that your hazard switch
may well be the next best place to look at for a simple fix, it pops
out quite easily and if you are feelin brave you can take the whole
thing apart and clean up the contacts as they could be slightly
corroded causing a high resistance which would give the flasher unit a
reduced voltage, possibly causing the buzzing you have described.

Another thing that makes me think that it could be your hazard switch
is that the headlight one is a similar design and that tends to die
after a while, not a nice occurance when you are bombing along a
country lane at 80+mph and then lose all your lights  :-)

And don't forget, just because you bought a new flasher unit from the
motor factor doesn't mean that it will work,  I went through about four
in a week from Halfords uk and then tried a 'proper' unipart one and
never had any problems after that!

Miniman