A Google Books search for "the nineteen tens" and "the nineteen teens" finds matches for both forms of the phrase going back many decades.
The earliest instance of "nineteen tens" in a Google Books search appears before the decade occurs, in H. G. Wells, "A Reply" to G. A. Paley's "Conciliatory Socialism" in The New Quarterly (1908):
Some of us, no doubt, will tend to constantly to a sceptical attitude towards the possibilities of conscious human co-operation, and some will tend habitually to a passionate faith. These will be the anti-Socialists and the Socialists respectively of the nineteen-tens and nineteen-twenties, the decades that lie before us.
Next come this mention in Cambridge Magazine (1913) [snippet]:
Aviation competitions. Foreign rivalry. The Kaiser. What a wonderful epitome of the Nineteen-Tens. However, it is not this alone, nor the adept way in which Mr. Bennett realises our speech-rhythms, which suggests this train of thought ; but the whole idea of the play.
And then this one from Ezra Pound, "'Dubliners' and Mr. Joyce" (1914), reprinted in Pavannes and Divisions (1918):
Mr. Joyce's more rigorous selection of the presented detail [in "Dubliners"] marks him, I think, as belonging to my own generation, that is, to the "nineteen-tens," not to the decade between "the 'nineties" and to-day.
The three earliest matches for "nineteen teens" begin a bit later—probably in 1923, though I couldn't verify the dates of the magazines where Google Books found these matches. From Keith's Magazine on Home Building (1923[?]) [combined snippets]:
True to the home-nesting instincts of men and birds and animals, this congenial couple planned vaguely someday to have this cosy roof-tree ; but those were the cheap living days of the nineteen-teens, the salary was all too easily spent, and apartment house walls had no yet grown oppressive. Thrift-times had been postponed ; then war-conditions broke, and they found themselves barely existing, the air-castle soaring out of sight, the rented walls contracting, and the landlord's exactions increasing.
From The New Yorker (1925[?]) [snippet]:
There was the Widow's, that farmhouse on Riverside Drive where you dined under the trees, or retired to a vine-shadowed porch which the young people of the nineteen teens found convenient for cocktails and embraces.
And from Amherst Graduates Quarterly (1930[?]):
The fifth decennial meeting of the International Congregational Council was held in Bournemouth, England, from July 1 to 8, 1930. Sixteen graduates of Amherst attended this notable gathering of Congregational ministers, the delegation including at least one man from each decade of classes from the eighteen-seventies to the nineteen-teens.
Perhaps most curiously, Ezra Pound (who accounts for the third-earliest instance of "nineteen-tens" above) refers to "nineteen teens" in a work published in 1938. From Ezra Pound, Guide to Kulchur (1938) [snippet]:
If I am introducing anybody to Kulchur, let 'em take the two phases, the nineteen teens, Gaudier, Wyndham L[ewis] and I as we were in Blast, and the next phase, the 1920's. The sorting out, the rappel a Vordre, and thirdly the new synthesis, the totalitarian.
As these instances (and later ones attaching themselves to the opposing camps) indicate, writers have long disagreed about how to spell out (and hence to pronounce) the decade of the 1910s; and as there is no arbiter to settle the matter, they will probably continue to do so.