
UPDATE
So the (very) good news: I called at exactly (the pips were still echoing on Radio 4) 9am on Tuesday, 11 June. When I got through on the phone the appointment booking system (used by the staff of Southwark Council to book appointments in) was not running, but it started to work (slowly) while I was on the phone. Although the system was super-slow, they were able to book my citizenship ceremony for 18 June! I went and I am now a UK citizen!

Not great News: I discovered when I was at Southwark Registry Office for the ceremony that the booking system is still not working properly.
Tip: Try calling right at 9am—you might be lucky (this might also work if you are calling to make an appointment for anything else as well—marriage, other nationality services, birth registration, etc). If you are applying for citizenship now, you might want to consider not requesting Southwark as the council you have the ceremony at (although they were great during the actual ceremony).