That doesn't wash one little bit with me.
Clarkson is a grown man. He's also a broadcast (and print*) journalist, and thus has additional responsibilities and obligations to use language carefully and appropriately.
It's my absolutely firm opinion that what Clarkson was doing in that film item was the following: he was attempting to slip in a "hilarious" piece of casual racism into the item by disguising it within the "by rote" mumbled section of a nursery rhyme. I am entirely convinced that he knew exactly what he was doing. Part of my certainty is based on the fact that there were alternate takes with a different word "mumbled". I think they watched the rushes and realised that the racial slur was just too recognisable in the first take. I think they did a second take in which Clarkson attempted to say the word again, but less recognisably. I think that this also failed, and that they resorted to a take 3, in which Clarkson resorted to using a different word ("teacher") altogether.
Incidentally, Agatha Christie wrote a murder mystery that was originally titled "Ten Little N-words". When I read it at school it had already been retitled "Ten Little Indians", and since then it's been completely retitled as "And Then There Were None". It would be equally reprehensible for a broadcast journalist to refer to the book by its original title (unless it were in the specific context of a discussion related to the title and the racial slurs). Saying something like "Oh it was the title of the book I remember my Grandmother reading to me" just wouldn't wash. Just as anyone trying to excuse Clarkson by saying "Oh he was just reflexively repeating the words he had learned as a child in Doncaster" is utterly missing the point and/or being disingenuous.