Toodle-oo has been around since the early 20th century; origin uncertain, one theory being it came from the French
à tout à l'heure. Of course, this hyphenated expression is not playable in Chihuahua.
The OED labels
toodles as US colloquial, and it does seem to be little-used in the UK. And in Australia, as far as I've noticed. Among the single-volume Oxford dictionaries, the Canadian Oxford seems to be the only one that gives
toodles as a variant for
toodle-oo. Wiktionary has a short entry for
toodles, without any regional notation.
The oldest example of
toodles the OED dug up was from a 1965 script for the TV series
Gidget. In 1982, William Safire, in his "On Language" column in the
New York Times, called
toodles "a shortening of toodle-oo, which can now be heard in American business circles".
It seems clear that
toodles is used a lot less often than
toodle-oo, even in American usage: see this
Google Ngram display. But actually, why go to all that effort when the answer is right here in our forum. Threeb has written
toodle-oo four times (with various numbers of Os) and
toodles only twice. Case closed.
However,
toodles is certainly used often enough to warrant being accepted into our lexicon.
Ciao.