It seems that the initial batch of invites that went out were mostly aimed at gamers who already played the PC, PS4, or Xbox One version of the game. If you already do you can head on over to Fortnite’s website where you will be asked if you already play the game, after which Epic will be able to locate your player data.
However if you don’t play the game, it seems that you will be put on a waitlist in which you will have no choice but to wait until you are invited. However if you already have a friend who managed to receive their invite, they will be able to invite you into the game thus skipping out on the queue.
If you don’t have that either, then we suppose you’ll just have to remain patient. It is unclear as to how long it will take for invites to fully reach all gamers, but presumably Epic is allowing batches in to prevent the servers from overloading. Once things have normalized we reckon players will have no problems getting into the game sans invite. Note that this is only the iOS version as the Android version of the game is expected to arrive later.