It sounds old fashioned and a bit twee. If someone said it, I'd assume they were just saying it for fun or as a little joke. In the early twentieth century, in the UK, people used to say 'toodleoo' or 'toodlepip' so it seems to be short for that. It's not really said these days though.

If you said that in the US, almost no one would know what you were trying to say.

I never hear "toodles" in the U.S. I have heard "ta-ta", a similar British expression, but only seldom.

That you can recognize the word is all you need to do with it. In Canada, when I was a kid we used to say "Toodles", "Toodle - oo", "Toodleootskies" and other such variations which I am sure we made up from some television show. Canada has a lot of English and American television so I can't say where it came from. "Later Gator" (short for "See you later, alligator") and all kinds of other fun kid sayings are tough for foreigners to follow, but are pretty much useless anyway apart from keeping children out of trouble. When I was in school trying to learn French, we used to make up words that sounded French to us, but I am sure no French person would understand. Thankfully we grew up! I am sure Japanese has lots of such fun sayings too.

"Toodles" is used in British English as an informal way to say goodbye. It is used in informal or friendly situations, so it could be used with friends or family. However it is a little old fanshioned and might seem a little strange if you use it in day to day life.