In the days of manual Morse, a telegram would originate at some office, be sent on the wire connecting that office to the nearest Western Union relay office, and then re-filed on a trunk wire that would take it as near as possible to it's destination, then back to a local or way wire to the destination office. It may have taken one or two more manual relays on trunk circuits to traverse the entire country. In your example,
From New York, in the pre-printer Morse days, the message would have probably been filed at a city office, sent to the main NY relay office on a city wire, then manually sent on a quadruplex thru circuit to "C" Chicago WU , manually relayed to another quad thru circuit to "DV" Denver WU Main, manually relayed to Pueblo WU relay, then put on the Thru wire to Durango, again manually relayed, where it would have been copied on a delivery blank and delivered to the recipient by a messenger as soon as it was received.
Telegrams had different classes of service. a "full rate" message would be sent immedately, and then put "in queue" in the sending files on the thru wires in the relay officesin order of receipt and moved as rapidly as possible towards destination. Wordage was kept low as those kinds of messages were the most costly, so the "paid for" word count probably would be only around ten words or so. People even used codes to reduce word count more if they could. Relaying a ten word message four or five times to get it across the U.S. might take most of a half day. Good Morse operators can send and copy a ten word message in less than a minute, so most of the time the message was in transit would be spent hanging on a hook "in queue" somewhere in the message files in the relay offices.
There were "DAY" and "NIGHT" letters too, that allowed up to 50 words or so, and were sent at a bulk rate, so were less costly for more words, but those were put back and handled at night or at times when the circuits came idle from handling the full rate traffic that took precedence over it. A "night" letter would be delivered usually within 24 hours or less.
Not the "lightning speed" of todays systems, but when it took days to cross the country by train, it wasn't too bad overall.
