Before I start I will say I know very little about cars so bear that in mind when I try to describe my issue.
My exhaust has fell off. So there's the bit of exhaust that protrudes the back bumper which is attached to a cylindrical box, that box has become unattached to the rest of the exhaust and it is now swinging about being held onto the the bottom of the car by a bolt.
It's going into the garage on Wednesday. Is it going to cost me a fortune? I can't seem to get a straight answer when Googling because I use phrases like "attached to cylindrical box".
Unless it is a complete all in one system, the main section of pipe from the engine joins the back box about 8 inches forwards of the box itself (ie towards the front of the car), the pipe from the backbox is flared to allow the pipes to join, the bit with the two slots in the top of the pic below. From your description, it sounds like the pipes have been seperated at this join, have you or anyone else hit a speed bump quite hard, or the edge of a kerb driving up/off?
Crawl under the car and see if the pipes have come apart, or if the pipe has snapped. If they have come apart, you should be able to push them back together. The pipes are held together with one of these, its an exhaust clamp, if that is missing, you'll need to get one from Halfords or a motor factors