According to the Wiki article, you can (supposedly) change your name, race, class and even birthsign using the console with
the ShowRaceMenu, ShowClassMenu or ShowBirthSignMenu commands and not reset stats back to their starting numbers -
PROVIDING that you somehow save your game from the console prior to clicking the DONE button.

However, all I get is "you cannot save at this time" whenever the various Menu screens are active. Is there some extra button
to press or click that I am missing somewhere? Or maybe the Unofficial Oblivion patch disables this?

When you're done editing your character, press the console button, and pause the game. The save option should be available now. Make a new save, don't overwrite an old one. Ater saving, load that save and you'll have your new character ready.

AHA! I wasn't pausing the game before trying to Save. Thank you very much

Whoops. The class and name changes work ok, but changing your race doesn't toggle the racial benefit/hazard routine.
My Khajiit - turned - Argonian can no longer see in the dark, but drowned when I wasn't paying attention to the breath meter.
The racial poison / disease resistance did not go into effect either.