I'm a professional translator looking for an English equivalent of a phrase that literally means 'the phones were hot' (–which doesn't really cut it in English!) I would actually associate a 'hot line' with a direct line between two people, such as the POTUS and Mr Putin. Phrases I have considered include: 'the phones never stopped ringing' (but this doesn't necessarily imply that they were actually constantly in use), and 'the phone system could barely cope' (but that doesn't have much of a ring to it [if you'll pardon the pun]). 'The phone lines were overloaded' is approaching what I'm looking for, but I'd like to find a phrase that is more graphic – more dramatic, in fact. I feel it is on the tip of my tongue, but it escapes me!
Additional info (as requested by a contributor): having thought more about the context, the phrase is used to describe what happened when managers within a huge company all suddenly reacted to a crisis by picking up the phone. Taking all the comments into consideration, the closest I can think of at this point is 'the phone lines were ablaze' (or 'overheating'), while a knowledgeable contributor has explained that 'congestion' is the correct technical term - which opens other avenues.