There's a button with a camera icon that lets me save a replay of a battle once it's over. The tooltip even says, "Save your replay to show off to your friends!" However, I don't see a way to actually play back those saved replays (let alone share them with others). I have looked everywhere and searched the War Metal Tyrant wikia, to no avail.
I guess I can actually answer this now. Players in the game chat claim that it only works for Facebook. Having checked all screens and buttons, I have not found a way to view replays or share them.