score:24
If you click on the "Apply" link and work your way through the umpteen questions to confirm eligibility (which only takes a minute or two), you'll eventually get through to the booking calendar page -- unfortunately linking this page directly from an external site doesn't work, but here's the status as of 2 October 2018:
As you can see, there are appointments available at every office on every day from the start of the 2-day booking horizon, with the exception of Newport and Glasgow on Saturday. I clicked through to the individual office pages for Thursday and found that, even just two days in advance, there are at least five different time slots available at each office.
Incidentally: in my experience with this service, the most difficult part of the whole process was producing an acceptable digital photo for upload. The online premium service requires a photo of the applicant's entire upper body, which they then crop down to a standard passport face size. Their automated photo checker is very fussy about things like uneven background tone, bright and uniform lighting, shadows, focus, and resolution. If I had to do it again I'd probably use a professional photographer for the photo.