recovery mode not available

Here's the story.

I want to upgrade my iMac.

We have an iMac from 2007 that is running OS X 5.6.8

The disc drive is broken because my 2yo stuffed too many discs into it and we do not have an external drive.

I have a time machine back up from my powerbook which was running snow leopard.

I want to recover the iMac with the timemachine saved from the powerbook.

I plug the external memory into the USB and use command R when I hear the sound of the start up. It doesn't go to the recovery screen but straight to sign in.

Then I read that maybe hold down the option key at start up so I try that. No dice.

Finally I see that maybe option-command-R could work. Still nothing.

Is there any hope for me to upgrade without taking my machine to the shop or Genius bar? Maybe through the internet?