I've got dual nationality (FVEY) and very specific clearances; in 2001 I was told to relinquish my other nationality by the UK (when I was working for the US State Department). I pointed out the tautology that this required me to formally renounce the Queen as Head of State! Fast forward to a couple of years ago when I had to undergo additional vetting for my latest appointment and my dual nationality was simply not raised as an issue.