Drove to London yesterday evening and stayed with a friend. Got up at 4.30 am and arrived at the Belgrano at 7.20. I was No 4, and was out of there, £78 poorer and with a new printed sheet for my licence, by 10.15...and back in Derbyshire by mid-afternoon.
So thanks for the advice, everyone. It was a long day, but much better than entrusting my log books to the post and waiting for weeks.
And after over 4 years and well over 300 instructing hours, I'm finally an unrestricted FI.