Just jumping in with some (hopefully helpful!) advice: when I first applied on a Sunday in late October, I immediately went to look for open Sopra Steria appointments, and found there were none. However, because I read on their website that new appointments are available every day, and it was already later in the evening, I decided to stay up until midnight, log in at 12am on the dot, and see if new appointments had been added. They had been, but it was really frustrating-- let's say that, on Sunday, 1 November, you could view Sopria Steria appointments up to Saturday, December 11. At 12:01am, when the date had rolled over to Monday, 2 November, you could suddenly see appointments for Sunday, December 12...but they were all already full--- this is because none of the centres I was hoping to book at were even open on Sundays! The systems just immediately shows all appointments as "full" for that day, even though the centre is actually closed (I know this because I actually live by one of these centres, and know it isn't open over the weekend). These weren't the exact dates, but I just wanted to let you know about this Sundays issue in case this happens to you! It's such a waste! Why they don't just skip over to Monday for those centres who aren't open is beyond me.
Anyways, I then decided to stay up until midnight again the next night and this next time, when the clock struck midnight and it was 3 November, I could see plenty of available dates for a Monday at several local centres. I had my pick of times. So my advice to you is to, if you can, maybe stay up late and try this if you haven't already?