(I'm a native US speaker). It means legitimate and honest. It means something is not a scam or a swindle. It is often used in connection with business dealings. Here's are some examples of usage. "Ponzi said that if you invested money with him, he would pay 50% interest in 45 days." "People should have known that wasn't on the up-and-up." "Why do I have to let them audit my books?" "They won't lend you money until they are sure everything is on the up-and-up." "I like this car dealer. I feel that everything there is completely on the up-and-up." I don't know if this expression is used in the UK.

If someone say they are “on the up and up” it means that they are honest or legal in American English. For example, “dad has always been on the up and up with them” means that he has always been truthful with them.

Dan has given a great answer. Just here to add that I have seen or heard this phrase before but it doesn't seem common to me. I know what it means, though. Perhaps from a TV show, so it is probably less common outside the US.

Generally, it means to be honest and reliable. “You can take his word for it. He has always been on the up and up.”

Google tells us that in American English "on the up and up" means "honest": https://www.google.com/search?q=define+%22on+the+up+and+up%22