I applied to Glasgow and got made an offer within a week or so. There were no interviews as such.
In an ideal world, all universities would interview all applicants. The reality is that staff nowadays are so busy that applications are mostly assessed on the UCAS form alone, and even the personal statement isnt universally used in the process.
Which does mean it's important that any prospective student should make sure that they find out as much about the university they're applying to themselves - particularly take any opportunity to visit and meet staff and existing students.
G