This may not strictly count because the time between the independently derived results may not have been long, but the significance of the result makes this example interesting and the two workers concerned certainly were unaware of each other's result.
The Russell Paradox was known to Cantor independently of Russell's announcement of the result and highly likely some years before Russell hit on it. See Jean van Heijenoort, "From Frege to Goedel: a source book in mathematical logic, 1879-1931" (1967) on page 114, where a letter from Cantor to Dedekind is reproduced. Cantor writes to Dedekind in 1899, two years before Russell announced his paradox:
"...If we start from the notion of a definite multiplicity (a system, a totality) of things, it is necessary, as I have discovered, to distinguish two kinds of multiplicities (by this I mean definite multiplicities).
For a multiplicity can be such that the assumption that all of its elements 'are together' leads to a contradition, so that it is impossible to conceive of the multiplicity as a unity, as 'one finished thing'. Such multiplicities I call absolutely infinite or inconsistent multiplicities.
As we can readily see, the 'totality of everything thinkable', for example, is such a multiplicity ..."
Cantor was aware that if you applied the Cantor slash argument to the set of all sets, a contradiction would follow. In this letter he shows he was aware of the contradiction inherent in the conception of this 'multiplicity as a unity, as one finished thing', and the Russell paradox in the words that Russell used was likewise found by Russell when he was seeking a flaw in Cantor's slash argument and applied it to set of all sets or, the 'totality of everything thinkable'. See the wikipedia article for the history of Russell's take on things.