Did you connect to an NTSC or PAL TV? I think the phone defaults to one or the other, and only switches to the selected one during booting.
My screen broke as well recently, and when starting the TV out wouldn't work, but after blindly pressing 'skip' on the initial settings screen from memory, the OS loaded, and the TV out started working.

Want a new screen? I have a new one I bought on ebay, but my N900 has died (EMMC failure) so the screen is of no use to me now... PM me if you're interested, I'm willing to sell at a slight discount.