I technically came to York via clearing, although York was also my firm choice, so I had already gone through the accommodation application process. My experience was that everything went smoothly, I didn't end up with my ideal accommodation, but my main areas (not being en-suite and being self-catered) were met (imo, the most important part is having nice flatmates). If I were you, I wouldn't worry about it.
Even if you don't get accommodation (which is unlikely), there are still a load of people with spare rooms looking for potential housemates around, and the places the Uni looks at (stuff like The Boulevard and Student Castle) are pretty nice, just quite expensive and not as sociable.