This topic covers a lot of ground.
Both AT&T and WU had carrier systems, or what we now call frequency division multiplex, that carried multiple telegraph circuits on a single voice channel, from the 1930s. WU also made extensive use of time division multiplex until the 1950s.
AT&T introduced TWX in 1931, using manual switchboards and DC loops from the switchboard to the subscriber. No doubt carrier systems were used to get inter-city trunks. This made the telegraph companies, W.U. and Postal, really mad, because it cut into their telegram business for the large business customers; but they didn't offer an effective competing service. Telex began in Europe about the same time, and differently in different countries. It was especially well developed in Germany, using a dialup network completely separate from the voice network, and using DC loops between the subscribers and the switching offices.
You could lease telegraph grade or voice grade private line networks from AT&T, and telegraph grade from W.U. These might be as simple as a line between two points, or a complete telegraph message or voice switching system with hundreds of points. Over short distances you could send any sort of DC signaling over a telegraph line; but over long distances there were regenerative repeaters in the line. These limited operation to Teletype standards of 5-unit code, 60 or 75 wpm. You could connect about anything you wanted to a voice grade private line (assuming you didn't transmit at too high a level) and in later years they could be conditioned (equalized) for data transmission using third-party modems. The SAGE air defense system had a lot to do with getting the Bell System into this environment of transmitting high speed data over leased lines. Unlike private lines, TWX and the voice switched network came under the "no foreign attachments" doctrine, meaning you couldn't legally connect anything of your own to the line; everything you needed was supplied by the telephone company.
In 1958 W.U. introduced Telex to the U.S., using switching equipment made by Siemens in Germany.
Some TWX subscribers were served using modems, which were single channels of frequency division multiplex carrier system. This was just for the convenience of the telephone company which might not have a suitable DC circuit going where the customer needed it. Or which might want to server several TWX subscribers over a single voice-grade line.
Circa 1959 the Bell System offered a modem, at the time called a "Digital Subset" for lease to subscribers. These were used, for instance, in the FAA ADIS weather data collection system, which operated a nationwide backbone circuit at 600 wpm. About the same time the Bell System set up a teletypewriter switching system for Delta Airlines, operating over the voice switched network using modems. The terminal stations had to be served out of #5 crossbar offices, so in some cases there were some long foreign-exchange lines.
During the same time frame the voice switched network was rapidly approaching a state of completion, in which customers could dial all long distance calls themselves and have the billing done automatically. (But as late as the 1970s I lived in a village where you could dial your own calls but then had to give your number to an operator for billing purposes.)
The billing problem was one reason TWX stayed so long with manual switchboards, while Telex was dialed in most of the world. Nearly all TWX and Telex calls are long distance calls. In the U.S. customers expect an itemized bill showing the place called and the number of minutes charged for. This was easy to do with manual switchboards. In Germany instead of an itemized bill the customer gets charged for a number of pulses. When a Telex connection is made a pulse counter associated with the caller's line is connected to a source of pulses, the pulse rate depending on the distance called.
As the voice switch network neared completion the Bell System decided to convert the whole TWX system to use dialup service over the voice network. The volume of TWX traffic was tiny compared with the rapidly growing volume of voice traffic, so it could be handled by the voice network without any substantial additions. There was also a desire to introduce ASCII in place of Baudot (Murray) code, but the substantial number of Baudot subscribers had to be accomodated. The end of all this was to develop modems to place at the customer locations and operate the Teletype machines over the switched voice network. There were some special arrangements to allow Baudot machines to call only other Baudot machines directly, and to go through speed/code converters to connect with ASCII machines, and vice versa.
These early TWX modems appear to be terribly overdesigned; but one must remember that the voice network was not all that good in the early 1960s. The modem design tried to guarantee a satisfactory 110 baud connection for almost all calls between any two points on the network.
It was also circa 1960 that Bell System people began to realize there was a substantial market for dialup data transmission between arbitrary kinds of business machines, even at the rates charged for voice calls. This was the result of rapid growth in online computer sytems, beyond the special cases like SAGE and airline reservation systems. Since Bell couldn't supply all the business machines, they altered the no-foreign-attachments rule by offering to lease dialup modems of various kinds. There were the 103 series at 150 baud, full duplex; the 201 series at 2000 baud synchronous, one way at a time, the 202 series for 1200 baud asynchronous, one way at a time. And there were modems for fax and Telautograph and other odd things.
W.U. tried over and over to get the government to force the Bell System out of the telegraph and data business. Government policy seems to have been to keep W.U. with exactly one foot in the grave at all times.