Uxie and Azelf Are Not Appearing in Nameless Cavern
Solution 1:
Actually it does indeed depend on the time set on your 3DS and having 3 max happiness pokes in your party. Uxie can only be caught between 20:00 and 21:00 hours. so between 8pm-9pm. Azelf can be caught between 21:00 and 03:59 or 9PM and 3:59AM and then mesprit the rest of the time. all the info is right here on this page on Serebii
http://www.serebii.net/omegarubyalphasapphire/legendary.shtml
I've confirmed from personal experience. everything on this page is true to the letter. have fun!
Solution 2:
They appear at different times. Uxie is from 8pm until 9pm, Azelf is from 9pm until 3:59am, and Mesprit you can find from 4am until 7:59pm. However, they will not appear if you have changed DS's or times and like with all legendaries within the game will not show up for 24 hours.
Solution 3:
I've found this thread, and there's an answer you may find useful.
You need the Eon Flute from Southern Island. Lati@s doesn't need to be in your party, just in your possession. Soar to Sootopolis and soar around until you find the red sparkle called "Nameless Cavern (I believe)". Land there and enter in. The Hoopa ring should be in there.
You need 3 max happiness Pokemon (I had a whole party of max happiness Pokemon when I caught Mesprit). I believe they are at certain times of day.
Hope this can help you, I haven't caught them yet so I don't know if this actually works.
Solution 4:
First, you must have at least three pokemon in you party with max happiness or friendship. I am unsure which. Second, you must find Nameless Cavern. It's a Mirage Spot southwest of mossdeep and northeast of sootopolis. Third, each will appear at a certain time, Mesprit is there from 4 am to 8 pm, Uxie is from 8 pm to 9 pm, and Azelf from 9 pm to 4 am. DO NOT TIME CHEAT! They will not spawn if you time cheat! I know from experience! The only way to get them to respawn if you time cheat is to beat the Elite 4, but I haven't tested this yet, I am going to tommorrow and post results.