As to Doctor Who, well, it's more soap opera than science fiction these days (and has been for a while), so it's no surprise that it's wildly popular in the states.
But the word descends directly from the French language; 30% of English is from the French, it's largest influence (another 30% is from latin, 30% from Germanic, and the rest from other sources). So it's not that we are choosing to 'spell things like the French': the word IS French, as are countless other words. It's our biggest source of vocab.
For those who first encounter the word "typo" indirectly as TamDenholm did, it's worth noting that a typo is not just any spelling mistake; it is a mis-spelling rendered in a type-set text and arising not through the author's ignorance of spelling but rather through a mis-operation of the typing apparatus (often arising through hasty typing). For elaboration, see http://en.wikipedia.org/wiki/Typographical_error.
If one believed (erroneously) that "spelling" were correctly rendered "speling", that would not be a typo but a spelling mistake. On the other hand, if one rendered it "speling" because of a finger slip on the second press of the "l" key then it would be a typo.
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Although it's interesting see the cross-pollenisation that seems to be happening linguistically thanks to media shifting both ways across the pond. Often used to be American imported media flowing into Britain with very few things leaving, but it seems like there's more American Doctor Who fans than British ones now.