I would interpret it slightly different to what John did.
I think your version is correct: ... if the telephone wires were cut, the rockets would be touched off to call for help from the artillery or to signal with
In other words, they would normally use the phone lines to call for help, or for various other messages. If they were cut, they would use the rockets. That might be to call for artillery help, or it might be for some other signal. Assumedly, the rockets had different colours, or some other way to encode a message.