Grammaticality of "On which hand it is worn should not make the slightest difference"
Solution 1:
I think that the quoted CGEL passage does not apply, because "On which hand it is worn should not make the slightest difference" does not contain a fused relative construction.
Rather than a relative construction, it seems to be an interrogative construction. ""On which hand it is worn" is not a noun phrase referring to the hand itself: instead, it means something like "The answer to the question 'On which hand is it worn?'", and as you noted in your question, it seems to internally have the structure of a clause (even though in the structure of the surrounding sentence, this clause occupies the place of the subject).
A set of similar example sentences that do involve a fused relative would I think be as follows:
"What it was worn on was his right hand." (= "[The thing which it was worn on] was his right hand.")
Seems to be grammatical, even if a bit awkward.*"On what it was worn was his right hand."
Ungrammatical, according to the CGEL quote. I think I agree with that judgement.