Here's a trick to make Google Music work every time: start playing a song, pause and play again. It's unfortunate that you have to use a workaround, but Google Music doesn't officially support Safari for iOS. The system requirements page says that it only supports "Google Chrome, Firefox, Safari, and Internet Explorer 7 and above" and that you need Adobe Flash.
It's interesting that this trick lets you play music on more than one device simultaneously. For example, I was able to play music on my iPad and iPhone at the same time, even if Google says that "music from your library cannot be played simultaneously on more than one device". The unofficial non-Flash interface is an unexpected loophole.