How do we go about naturalizing my American wife subject to this exception?
This particular paragraph is poorly worded, sorry about that. Most of the time the BBC is superb. There's some exceptions to the 3 year rule if she's already settled in the UK and you were pursuing commitments abroad. Or if you have been working in "Crown Service", there's something to be done sometimes. Otherwise, it's a no. Sorry

*BUT* to answer your question how to go about applying for an exceptional: that's quite easy! Just download the AN1 form and have your wife fill it out. Then include an attachment that justifies your applying for a concession and send it to the British Consulate in charge of your area. Hopefully, you would print out a copy of the BBC page and include it with your wife's application so they would know where you are coming from. You will receive an answer in about 3 - 4 months in the straight-forward case. I wouldn't advise it because it's a waste of money, but it does answer your question.
In the normal case, you would go to
http://www.londonelegance.com/transpondia/fcoverview and follow column 3. As shown, your wife is eligible to enter the fast track because you have been married 4+ years abroad...
That's an opinion based upon my own understanding of the rules; and I should hope that others will provide alternate interpretations.