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30 min for bounced out and usually 3 days for a win.
That being said the Royal does have a timer so if you lose near the end of the time you will not see it again until it would normally start again ie 3 days.
Also 3 days is generally how it works. Sometimes patches, group changes, or something we don't know about can lengthen or shorten the time between the next Royal. It is rare but does happen.
Also any 1000 or 1500 gold weekend versions end at around at 9am UTC, 5am US east coast. This does not end the Royal itself but it reverts back to 500 gold. Catches a lot of people.
30 minutes refresh after being bounced out of the challenge and a three day wait are the official times @Isaac_KC and that is what I get in my game.
There are many complaints from people that say their Royale constantly appears between levels though.
Very briefly my Royale did not show the animation of the 100 players being added, also I had a version that after I had been bounced out didn't start again until the next day.
Difference like this indicates that King were/are tinkering with the feature and there will more than likely be a few different versions of the Royale out there.