As best I can tell, the Oaks did sport a blue/white uniform by ca. 1912. There's a few different tobacco cards from that era that depict such a uniform. See below.
By the latter portion of the decade, the Oaks used a different uniform than the one below. I wondered if this one perhaps was the early green/yellow one they may have also worn? Then I found this 1918 movie footage of the Oaks and their opening day festivities:
https://www.youtube.com/watch?v=7u-uQBv7H1Y
Obviously, this footage has been colorized. But, however they colorize these old movies ... they made these uniforms blue/white, too.
So, who knows?
It's a cool movie clip either way and that first uniform is totally sweet.