It is possible to make it past the waterfall at the end of Zora's Domain without Zelda's Lullaby, and some of these methods are faster than the Lullaby scene to slow the water anyway.
Grab the cucco at the beginning of the river and bring it all the way up to the waterfall. Go to the very left of the closest ledge to the waterfall. Now face the wall sligtly left of the water fall and jump at it. Hug the wall and go a little bit right until you get in the tunnel.
As adult simply using the hover boots like the cucco method will do.
Both the cucco and hover boots methods can be done from the right side, however it's a bit harder and you have to grab the ledge.
It is also possible to get past the waterfall to Zora's Domain with a megaisdehop.