I'm making some changes to the database this morning to see if that helps (the changes are being made now, so it may be a little slow for a few minutes, but no downtime).
I'll also increase the scope of the slow query log to see if there is anything dragging the performance of the site down. In the early hours of Sat morning (around 3-6am GMT) there may be some slowness due to the backup script running, but it shouldn't last this long.
Hopefully these changes will improve the situation, but it if happens again tomorrow just let me know and I'll delve into the new logs.