how about a sprite animation for "shine" ?
Basically, all you have to do is, to trigger the animated sprites in correct order.
The other way I would try would be writing a custom shader that uses another textue input for a gradient, a float value for completion percentage and simply a screen space uvw mapping. It's not that difficult but I'm not entirely sure if it's needed for this case as I can see from the example video the sprites' shine is also being triggered separately as well.
Does it make sense?