As far as I'm aware each case is judged on it's merits and, provided the "blip" is declared and of course not terrorism-related, you might be ok, although DUI is obviously very serious in light of the responsibility you'd have as an ATCO.
When I joined the company the check was a simple CTC procedure - done by MoD staff, which involves amongst other things checking your name and details against the nationally held terrorist database. I think now, in addition to this, there are the new checks relating to the new security acts that go a little further.
Some criminal convictions preclude any form of airside access - I'm not sure of which etc so I hope someone else comes along who is rather more help
Have you already applied and reached the third stage I wonder?
Or are you asking before you put in any sort of application?