It's possible that before playing a level, some of your characters may not be loading as expected (downloading message appears on screen).
In the event of re-installing the app, all unlocked characters will be downloaded. The characters are downloaded again in random order.
You may think that the downloading is not taking place since the progress bar is not moving, however the downloading is happening in the background. If your internet connectivity is not strong enough, downloading all characters and costumes may take a little while. Be patient, your characters will be right there in a moment!
If possible, try to avoid uninstalling the game and instead force close the app and restart it to prevent having to download all this data again.