The first automatic radio telephone service
On March 1, 1948 the first fully automatic radiotelephone service began operating in Richmond, Indiana, eliminating the operator to place most calls. [McDonald] The Richmond Radiotelephone Company bested the Bell System by 16 years. AT&T didn't provide automated dialing for most mobiles until 1964, lagging behind automatic switching for wireless as they had done with landline telephony. (As an aside, the Bell System did not retire their last cord switchboard until 1978.) Most systems, though, RCCs included, still operated manually until the 1960s.
Some claim the Swedish Telecommunications Administration's S. Lauhrén designed the world's first automatic mobile telephone system, with a Stockholm trial starting in 1951. [http://www.telemuseum.se, link, now dead] I've found no literature to support this. Anders Lindeberg of the Swedish Museum of Science and Technology points out the text at the link I provide above is "a summary from an article in the yearbook 'Daedalus' (1991) for the Swedish Museum of Science and Technology http://www.tekmu.se/, link now dead]." He goes on to say, "The Swedish original article is much more extensive than the summary" and that "The Mobile Phone Book" by John Meurling and Richard Jeans, ISBN 0-9524031-02 published by Communications Week International, London in 1994 does briefly describe the "MTL" from 1951. But, again, nothing contradicts my contention that Richmond Telephone was first with automatic dialing.
On July 1, 1948 the Bell System unveiled the transistor, a joint invention of Bell Laboratories scientists William Shockley, John Bardeen, and Walter Brattain. It would revolutionize every aspect of the telephone industry and all of communications. One engineer remarked, "Asking us to predict what transistors will do is like asking the man who first put wheels on an ox cart to foresee the automobile, the wristwatch, or the high speed generator." Sensitive, bulky, high current drawing radios with tubes would be replaced over the next ten to fifteen years with rugged, miniature, low drain units. For the late 1940s and most of the 1950s, however, most radios would still rely on tubes, as the photograph below illustrates, a typical radio-telephone of the time.

Visit the Telecommunication Museum of Sweden!http://www.telemuseum.se/historia/mobtel/mobtfn_2e.html (link now dead)
Let's go to Sweden to read about a typical radio-telephone unit, something similar to American installations:
"It was in the mid-1950's that the first phone-equipped cars took to the road. This was in Stockholm - home of Ericsson's corporate headquarters - and the first users were a doctor-on-call and a bank-on-wheels. The apparatus consisted of receiver, transmitter and logic unit mounted in the boot of the car, with the dial and handset fixed to a board hanging over the back of the front seat. It was like driving around with a complete telephone station in the car. With all the functions of an ordinary telephone, the telephone was powered by the car battery. Rumour has it that the equipment devoured so much power that you were only able to make two calls - the second one to ask the garage to send a breakdown truck to tow away you, your car and your flat battery. . . These first carphones were just too heavy and cumbersome - and too expensive to use - for more than a handful of subscribers. It was not until the mid-1960's that new equipment using transistors were brought onto the market.Weighing a lot less and drawing not nearly so much power, mobile phones now left plenty of room in the boot - but you still needed a car to be able to move them around."
The above paragraph was taken from: http://www.ericsson.com/Connexion/connexion1-94/hist.html Ericsson has since removed this information from their website. You might try Alexa.com to do a Wayback Machine search.
In 1953 the Bell System's Kenneth Bullington wrote an article entitled, "Frequency Economy in Mobile Radio Bands." [Bullington] It appeared in the widely read Bell System Technical Journal. For perhaps the first time in a publicly distributed paper, the 21 page article hinted at, although obliquely, cellular radio principles.
----------------------
Resources
Bullington, Kenneth "Frequency Economy in Mobile Radio Bands." Bell System Technical Journal, January 1953, Volume 32: 42 et. seq.
Douglas, V.A. "The MJ Mobile Radio Telephone System." Bell Laboratories Record December, 1964: 383
Gibson, Stephen W., Cellular Mobile Radiotelephones. Englewood Cliff: Prentice Hall, 1987. 8
McDonald, Ramsey "'Dial Direct'" Automatic Radiotelephone System. IRE Transactions on Vehicle Communications July, 1958: 80 (back to text) As a courtesy to researchers I have scanned this article for you to download and review. These are very large files but they are readable and with some work will be decent for OCR. The first image is the title page for the IRE Transactions publication. The article starts at page 80:
http://www.privateline.com/IRE/IREfrontpiece.jpg
http://www.privateline.com/IRE/page80.jpg
http://www.privateline.com/IRE/page81.jpg
http://www.privateline.com.com/IRE/page82.jpg
http://www.privateline.com.com/IRE/page83.jpg
http://www.privateline.com.com/IRE/page84.jpg
http://www.privateline.com.com/IRE/page85.jpg
[McDonald2] ibid. 84
O'Brien, James "Final Tests Begin for Mobile Telephone System." Bell Laboratories Record July/August, 1978: 171
[SRI1] David Roessner, Robert Carr, Irwin Feller, Michael McGeary, and Nils Newman, "The Role of NSF's Support of Engineering in Enabling Technological Innovation: Phase II Final report to the National Science Foundation. Arlington, VA: SRI International, 1998.
[SRI2] ibid.
Young, W.R. "Advanced Mobile Phone Service: Introduction, Background, and Objectives." Bell System Technical Journal January, 1979: 7 (back to text) Messrs. Carr. Feller, McGeary, and Newman, of SRI, supra, cite the original memo describing cellular as follows: "Mobile Telephony -- Wide Area Coverage" Bell Laboratories Technical Memorandum, December 11, 1947.
[Discussion] Some might say conventional mobile telephones already employ frequency reuse since the same frequencies are used in radio-telephone service some distance away, in other cities perhaps seventy miles or more distant. Broadcast radio and television stations use this same approach to prevent interference, where the same frequencies are used throughout the country and where each station is separated by distance or space. In cellular, though, frequency reuse goes on within the fixed wide area of a cellular carrier, as part of an overall operating system. Within the coverage area of an AM or FM radio station, by comparison, no other station can use the frequency of that station. And there is no connection between other stations to act as a network.