The main link in the chain was the wirephoto, developed in 1935.
Vietnam-era news photographers shot 35mm black-and-white film, for the most part. Though this could be developed and printed in the field, in practice either the photographers or their rolls of exposed film went back to Saigon. In well-equipped darkrooms there, selected images would be printed using an enlarger. Pros can go from undeveloped film to finished prints in less than an hour.
Finished prints would then be mounted on the wirephoto machine and sent back to AP or UPI headquarters, from which they were retransmitted to subscribing newspapers. A few large newspapers had their own Vietnam photographers, but it seems likely they arranged to use the wire service machines rather than having their own machine in Saigon. Europe and the US were linked by a leased circuit, not subject to the vagaries of long-distance telephone switching circuits, by 1963. I’m not clear if there was a similar arrangement for Vietnam, but by 1966 it certainly would have been justified. Sending an image generally took about eight minutes.
Color photos could also be sent, by using color filters—just as were used in making four-color halftones in the printing processes of the time—to send three or four images that would then be recombined by the receiving newspapers. Very few newspapers of the era used color photos in regular daily editions, though, and the wirephoto process was rather crude resolution, even lower than the standard 65 dots per inch that newspapers used. Color newsmagazines—Life, Look, Time, Newsweek—instead flew the exposed color film back to the US for processing and four-color process printing.
Television coverage still meant 16mm film, which could easily be shot in the field, and developed in a Saigon lab. After being cut into a short edited piece at the Saigon bureau, this would be projected onto a special telecine television camera, which could transmit via satellite or coaxial cable to New York videotape machines, for later transmission on the evening news. Portable color electronic cameras weren’t developed until the mid-1970s. Remember that in 1969 it took the resources of the space program to develop a television camera small enough to take to the moon, even for black-and-white.
As for the newspapers making copies, the received wirephotos were handled the same way as photographs handed in by local photographers. The original continuous-tone photo was rephotographed onto orthographic (high-contrast) film that was sandwiched with a screen of closely spaced perpendicular lines. Where the image was bright, light diffused (spread out) and created large dots. Where the image was dark, small dots were created. This halftoned negative was then stripped into a form for photo-offset platemaking (rare for newspapers of the era), or was used for a photoengraving process that created a plate with raised dots for traditional letterpress or newspaper printing. Cut to the obligatory shot of the papers rolling off the presses; fade to the bicycling paperboy throwing the folded papers into suburban shrubbery.