Depends on the subject. Any high energy physics place is "Do Part III, get a distinction, then we'll talk" or perhaps an extremely high mark from the 4th year of a few select other unis. For fluids you can get a place with a merit, but they say "A distinction will certainly get you a place, so aim for that" and then accept a number of people with merits.
Since everyone doing Part III is given an offer if they apply, the offer is always "Distinction, even then it's not certain" so that they can drop the offer if circumstances warrant it, but aren't bound by offering 10 people merits for 2 places and everyone getting merits.