I am relatively new to the game and just defeated the Ender Dragon for the first time, but no egg appeared on the tower.  It appears this issue was reported previously (MCPE-28864), and has supposedly been resolved. 

I defeated the dragon in Peaceful mode, and went through the End Gateway before looking for the egg.  After going to the end city I remembered I hadn't gotten the egg, but when I came back it wasn't there.  I looked at the world spawn point, as well as all around the Exit Portal.