This may now be possible using IFTTT, because Alexa alarm and timer triggers are now supported. However, you have to tie the Echo's alarm or timer going off to making music play in some other app or web service. So you could have the alarm trigger music to start playing in last.fm, Google Drive or Soundcloud, for example, and it would pipe to the Echo's speaker if Bluetooth is enabled on the device where the music is playing (assuming the device is within Bluetooth range). Having the "Happy Birthday" in between the two things is more questionable.
I think Y's solution is much more elegant.