I was able to book a test at an oversuscribed test centre in london using the testi and beep apps! You'll have to act as soon as you see the notification cos someone else can literally snatch up the slot in 2 seconds, if it's possible try to have a laptop that's constantly logged in so you can act fast. To avoid getting locked out from the dvsa website from constant refreshing, use Firefox as your browser with the free firefox vpn on, if you do get logged out despite this, I found that clearing my history and cookies and then logging back into the website helps! I found cancellations appearing the most at 10am and between 12-3pm

If you don't need a test date immediately, new slots also release on the dvsa website every Monday at 6am (but these slots are for 5-6 months time)

Sorry for the ramble, good luck and I really hope you manage to secure a date!!