I haven’t found any other sources that agree with UD.
These two align with the Cockney-rhyming-slang (CRS) origin of “Taking the Mickey Bliss”; the latter cites a first appearance in print to the 1930s.
This makes the UD article (Bing cache, since work web-blockers block that kick) problematic, in that is speculates that “Britons have been using this figure of speech for decades, if not centuries.”
Discussion of UD article vs other sources: http://www.alphadictionary.com/bb/viewtopic.php?f=10&t=6447 (Trigger warning: people referencing ngrams and other linguistic terms).
This thread cites Partridges Dictionary of Slang and Unconventional Language - a thoroughly enjoyable read, btw - which also holds to the CRS origin.
So, my verdict? Anybody trying to claim “That’s racist!” is being disingenuous, or trying to slip you a mickey so you won’t notice.