As a supplement to Jon Hanna's answer, here are some very early instances of phone that appear in connection with telephone (though not necessarily as an abbreviated form of that word) from newspapers collected in the Library of Congress database. Most of these instances are either jocular or purposely generic in order to cover such recent inventions as megaphones, microphones, aurophones, and phonographs, in addition to telephones. But a couple of the instances seem to be specific to telephones.
Instances of 'phone' as a short form of 'telephone', 1878–1879
I found two serious and specific examples in which phone seems to serve as shortened form of telephone. From "Local Matters," in the [Richmond, Virginia] Daily Dispatch (March 16, 1878):
The telephone can be used with equal convenience at the mouth, as a speaking trumpet to transmit your words to the person with whom you are in conversation, or as a ear trumpet to hear what he says. For one person two of these 'phones are provided. One is usually held at the ear; the other at the mouth. For greater distinctness in hearing on can be applied to each ear.
Messrs. Kates and Paynter conducted the experiment at this end o the line, while Mr. N. C. Pamplin ("old Nick"), the manager of the telegraph office at Norfolk, operated down there. Nearly every word uttered at Norfolk could be heard with a 'phone at your ear.
In the singing it will be understood that each singer put a 'phone to his mouth and each hearer put one or more to his ear. That is the great advantage of this particular telephone—the 'phone can be applied with equal facility to mouth or ear.
And from "Local News," in the [Barton, Vermont] Orleans County Monitor (May 19, 1879):
C. F. Davis has just put up a telephone, which connects his shop with his residence. Conversation is easily carried on between the two places. He runs a small copper wire between the two places. The wire is attached at each end to the vibrating discs of the phone.
Jokey or generic instances of 'phone' in association with 'telephone', 1878–1879
Instances of phone used for humorous purposes or as a generic term for "fancy invention involving sound and wiring" (or both) are more common than straightforward instances of phone as a short form of telephone. My search of the Library Congress's Chronicling America newspaper databases produced six unique matches of this type from the years 1878–1879. From "Rocking the Cradle," in the [Diamond City, Montana] Rocky Mountain Husbandman (June 20, 1878):
A few days since the Burlington [Iowa] Hawkeye propounded this conundrum:
"Couldn't Mr. Edison invent some kind of phone or graph that would be acted on by the cry of an infant in the night so it would instantly grasp the cradle and rock like the tossing billows on the ocean's breast. ... Speak up, Mr. Edison."
And lo! Mr. Edison has gone and done it. He has a cradle with a telephone, a battery and a box containing magnets, and connected with a lever. The baby's cries are received by the telephone, which, operating through the battery and magnets, sets in motion the lever which rocks the cradle. When the baby stops crying the cradle ceases to rock.—Dunellen [New Jersey] Rock
From the Los Angeles [California] Herald (July 4, 1878):
The "Tasteograph" and "Feelophone" are said to be the two latest inventions. The latter will doubtless prove the most popular of all. It is intended to multiply and magnify pleasurable feelings or sensations to any requited extent and a thousand fold, if desired. The "Tasteograph," to the gourmand and bibulous admirer of Roederer or old Burgundy, will be indispensable to the augmentation of the delights of the papillæ. Bring on your 'phones!
From the [St. George, Utah] Union (July 12, 1878), reprinted from the Burlington [Iowa] Hawkeye:
A "Phone" Worth Something
An intelligent farmer living in Des Moines county, has invented a henophone, modeled on the principle of the telephone, by which one old reliable hen occupying a central office in the henery sits on all the nests about the establishment, leaving the other fowls free to lay eggs, scratch and cackle.
From "Thomas Edison," in the [Carson City, Nevada] Morning Appeal (August 4, 1878):
"He [Edison] intended to stop here [in Reno] for a day for the purpose of seeing Frank Bell in relation to the introduction of his various "phones" on this division of the Western Union line, but Mr. Bell, being West on business, informed him by telegraph of his whereabouts, and the distinguished gentleman continued on his trip.
From "The Crowning Discovery," an advertisement in the Newberry [South Carolina] Herald (February 26, 1879) and in many other newspapers over the next several months:
All the "phones" of this phonetic age are surpassed in practical benefit to mankind, by the discovery of Allan's Anti-Fat, the great and only known remedy for obesity, or corpulency. It produces no weakness or other unpleasant or injurious effect, its action being simply confined to regulating digestion, and preventing an undue assimilation of the carbonaceous, or flesh-producing elements of the food.
From the Worthington [Minnesota] Advance (September 25, 1879):
One of our subscribers wants us to tell him about the telephone. We never tell a phone. Never told a phone in our life. We may say, however, that the telephone was invented by Edison. He did it with his little hatchet.
These examples underscore Jon Hanna's point that phone came into use—both as a short form of the word telephone and as a generic way of referring to "all those newfangled inventions that have phone in their names"—from a very early date after the invention of the telephone. Although telephone was more common than phone in published English for many years, and although it remains quite popular today, the short form phone is almost as old as the long form telephone.
Among other things, the excerpts include one example from March 1878 in which 'phone (with an apostrophe) appears as an abbreviated form of telephone, and one example from May 1879 in which phone (without an apostrophe) appears as an abbreviated form of telephone.