As far as attestation of 'Soviet' goes, OED (paywalled) abdicates by attesting the noun sense A1a from 1917, and annotating the sense to acknowledge earlier uses (bolding mine):
In the U.S.S.R.: one of a number of elected councils which operated at all levels of government, having legislative and executive functions.
The term was also applied to various revolutionary councils set up prior to the establishment of socialist rule in 1917.
The OED etymology given is "< Russian sovét council"; observe the diacritic, which represents OED's "more 'scholarly'" (see the third paragraph of the "Headwords and the selection of entries" section in the "Preface to the Third Edition of the OED") transliterative romanization of the vowel ye ("Name of letter") in the cyrillic original.
A newspaper search for earlier attestation of 'Soviet' did uncover this from a
"PROCLAMATION" attributed to "Michael Obrenowitsch, Prince of the Servians" in The Times (London), 01 May 1841 (paywalled, bolding mine):
The political events which troubled the tranquility of our country in the past year occasioned several members of the Soviet to lay down their appointments, partly because the people had risen against them, and partly of their own accord.
(The history of this proclamation is, of course, interesting in its own right, but omitted here.)
Serbian, while very similar to Russian, was not identical, and the value of the transliterated use of "Soviet" in The Times amounts only to confirmation that the term 'Soviet' was expected, in context, to signify a type of Slavic "council, board" to an educated, English-speaking audience as early as sometime prior to May, 1841.
Additionally, a search in the Google Books corpus (via HathiTrust) turned up this phonetic transliteration (I assume) of the Russian word 'совет' in the 1833 Ivan Vejeeghen; or, Life in Russia, by Thaddeus Bulgarin:
...who have lately come of age and borrowed three hundred thousand roubles in the Opekoonsky Sovyet?*
....
* The name of the bank belonging to the Foundling-hospital.
Later, in the first decade of the 20th century, I found uses of 'Soviet' in 09 May 1905 The Indianapolis News, Indianapolis, Indiana (pages 4 and 6, emphasis mine). The first is a syndicated article distributed by the Associated Press; the second is a semi-humorous 'blurb':
NEW COUNCIL FOR RUSSIA IS FORMING
Character of the Branch of the Government Provided by Czar's Rescript is Defined
"ZEMSKY SOVIET" IS PLANNED
Membership Will Be 500 Men to Serve as Intermediary Between People and Ruler.
ST. PETERSBURG, May 9, 3:07 a.m. — A report defining the form and character of the new branch of the Government to be created in accordance with the imperial rescript of March 3 is current in official circles here, according to which the fruits of the Bouligin Commission have taken the form of a resolution for the creation of a "Zemsky Soviet" or council of Zemstvos, elected indirectly through the Zemstvos, to serve as a connecting link between the Emperor and people.
....
Though such a Zemsky Soviet will fail to satisfy the Liberals, who are calling for a full plunge into constitutionalism, universal suffrary, a responsible ministry, etc., ....
[Page 6:]
The latest dispatches announce that it is a Zemsky Soviet which is to be created in Russia. Presumably it was discovered that a Zemsky Sober was impracticable.
From an editorial titled "Russia's Awakening* in the 14 May 1905 The Indianapolis News:
....
Now comes also the entering wedge for constitutional government. The Bouligin Commission appointed some months ago by the Czar has reported a scheme for a popular assembly of an advisory character, to be known as the Zemsky Soviet.
Another syndicated article, datelined St. Petersburg, 18 May 1905, was picked up by two newspapers in the Newspapers Extra corpus:
WILL HE MAKE GOOD.
It is up to the Czar to Keep His Promise.
...All St. Petersburg and Russia in general are waiting today to see if the czar will make good his promise of last March to form a new branch of the government to serve as a connecting link between the emperor and the people. The new branch will be called the "Zemsky Soviet" and today has been set as the date of the proclamation putting it into effect.
These were followed by more widespread appearances in May and late October, 1905, in short syndicated articles documenting the creation of the Zemsky Soviet. One of the May articles, in the New-York Tribune, seemed to have been editorially altered to use the 'Sovyet' variation of 'Soviet'.
Of the October articles, another syndicated (AP) article appearing in the 26 Oct 1905 Leader-Telegram, Eau Claire, Wisconsin and other newspapers, describes 'soviet' as the "old name":
....
It is understood an edict formally establishing a ministerial cabinet which will bear the old name of "soviet," or council, will be published....
Substantially the same article appeared on 26 Oct 1905 in The Billings Gazette; that article had been locally edited to delete "old" before "name".
At about the same time (Sep-Nov 1905), articles in the popular press of both the UK and the US made mention of the Soviet Siezd (Council of Associations of Naptha Producers; Siezd is also spelled Sieza in these reports).
The tendency of these (and, presumably, other) mentions of 'Soviet' in the popular press was to firmly entrench the term as the English for a "Slavic council or governing body". At the same time, the occasional use of the orthographic form 'Sovyet' suggests the word was at least perceived by English speakers as having been adopted from Russian or another Slavic language.
Altogether, early uses, including those I managed to uncover in the first half of the 19th century and the first decade of the 20th, along with the general sense that the more usual phonetic representation in roman characters was -ie-, may have led to -ie- "winning out" over the simpler -e- for transliteration of the cyrillic сове́т. As commenter @sumelic made clear,
Russian speakers don't hear "ye" as in "yet" in this word, although English speakers do; to a Russian speaker, it's about the "v" being "soft" or "hard".
