This is the busiest time of the year for new customers as lots of people rush to have it in place for Christmas. Unfortunately, as with every single service provider, current customers come behind new ones every time. If you want to get a rush on it, perhaps asking to cancel the service might bump you up the list.
However, pretty much every issue bar the box itself failing should be an easy DIY fix. From what you're said so far, I'm guessing you have Sky multi-room, with feeds from the dish going to Sky boxes in the two rooms. Correct so far?
As GNR asks, is the box that isn't working powering on and giving a no-signal error? If not, the box is the problem.
Have you tried swapping the two boxes? If the box that wasn't working in one room doesn't work in the other, then the problem is the box. So keep the working box in the room you'll use most.
If both boxes work in one room, but not in the other, then there is a problem between the dish and the box. First, check all connections into the back of the box. Sky+ boxes will have two screw-on connectors feeding from the dish. Check that these aren't loose, and the cable is seated securely in the connector. Undo and remake these connectors if required.
Next, check the connections to the LNB (little box on the arm sticking out from the dish). Are these connectors still secure? Swap the connectors around to different sockets and see if that makes a difference.
You could get a long length of sat cable and put a plug on each end, attach that directly between the dish and the box that isn't getting a signal to bypass the existing cable. If that works, you know the cable is the problem. If after checking the connections on that cable doesn't yield results, easiest solution is to run a new cable.