What a freaking nightmare to sort this out. Still not 100% sure what the issue is, hosting provider is not particularly communicative.
I've setup CloudFlare for our site so you might experience some interruption while things propagate around the world. It's supposed to help with speed (doubling it) and preventing load attacks. We shall see.
I've implemented a new caching system that should help but the slowdown was due to a weekly backup of the system that takes a while to complete. Once it's done the system returns to normal performance. Backups happen on a weekly basis at 2AM EST so expect a few hours of slowdown while it finishes.