Edit: Updated Answer
After more research, I verified that the OED does observe the distinction between initialisms and acronyms, invited by the distinct senses (in the OED) of 'initialism' and 'acronym' (as abbreviations pronounced as individual letters in the former, and abbreviations pronounced as words in the latter), in its Etymology notes. I respect that distinction in this answer.
That verity enabled the simplification of research into the earliest acronyms appearing in the OED. In the following historical breakdown, I only include words with clear, unequivocated acronymic etymologies. Thus,
Of words originating in English between c1000 and 1600, no acronyms appear in the OED.
Of words originating in English between 1600 and 1800, the noun 'abjad' is undoubtedly an acronym. It derives from the Arabic word 'abjad' (also acronymic), denoting the Arabic alphabet: the word is an acronymic abbreviation formed from the initial letters of the names of the first four letters of the Arabic alphabet. The first textual evidence of its use appeared in 1793:
1793 C. F. Greville Brit. India Analyzed I. 87 The first letter of the month Ahmudee is Alif, which, in the Abjud, stands for 1.
Thus, 'abjad' has the honor of being the earliest acronym in English documented by the OED. No other serious contenders for the earliest acronym were to be found in that period (1600-1800).
Being curious, I continued the exercise.
Of words originating between 1800-1900, there are several acronyms. In historical order, those are 'Tanach' (1835), 'SCOTUS' (1879), 'OUDS' (1885), 'AWOL' (1894), and 'POTUS' (1895).
A further search for acronymic words originating between 1900-2000 returned 287 results. I did not visit each entry (as I did for the earlier, more managable quantities of results) to verify that the reference to 'acronym' in the Etymology was clear and unequivocal, and to verify the date of the earliest acronymic use of the word.
The question invites argument: "the earliest example" in the question could resolve as the date the abbreviation was first pronounced as a word, or the first recorded appearance of the abbreviation later to be pronounced as a word. If the former, how is that date to be documented?
Anyway, I'll throw a word into the ring:
AWOL, n. and adj.
1894 Rep. National Home for Disabled Volunteer Soldiers in Misc. Documents House of Representatives 1894-5 (1895) 431 November 25: Patient left the ward this date without leave. December 5: Patient returned from A. W. O. L.
Note that the OED distinguishs 'acronym' and 'initialism' on the basis of whether or not each letter or part is pronounced separately: for initialisms each letter or part is pronounced separately; acronyms, on the other hand, may have each letter or part pronounced separately or the whole may be pronounced as a single word.
Thus, my rather shaky assumption is that where the OED gives an etymology as "acronym" or "initialism", it maintains that distinction. This assumption is supported by the etymology given for 'awol':
Etymology: As noun, acronym < the initial letters of absence without leave ....
In early use sometimes an initialism; compare quot. 1949 at sense B. 2, and also:
1921 Outing June 137/1, I was surprised to find one day that unless I left the following morning to rejoin my regiment I would be an ‘a-w-o-l’.
1957 B. Evans & C. Evans Dict. Contemp. Amer. Usage 9/2 awol..in World War I..was pronounced as four letters; in World War II, it was pronounced as a word.
(op. cit., italics mine)
Rather than helping, however, the OED's distinction leaves the question of when an initialism first becomes an acronym up in the air. Presumably, the question is unresolved because documentation will rarely answer it, and because the question is extraneous to the OED's primary concern with when the acronym/initialism was first recognizably used as a noun (adjective, whatever) as documented in textual records.
For somewhat hesitant and possibly unreliable support of my claim that awol was the first or among the first acronyms, as well as a disquisition emphasizing the distinction between acronyms and initialisms, see "Acronyms that became words".