I came from a maths degree beforehand. In my class of 20 there were 7 or 8 of us from maths, plus a handful of engineers. But there was a variety of other backgrounds, so the course I was on at least, is open to many.
I recently took a job in insurance – so I don't consider it strictly OR – but it's not far off, and the qualification definitely helped show my skills.
I finished my dissertation in September and didn't look for any jobs until after I'd finished. Within a month, out of 3 interviews I had 2 offers and the chance of a 2nd interview so I think the prospects are reasonable.
I know Southampton, Warwick and Lancaster all have good reputations for their OR departments.