HDMI is typically not available on boot, and by that I don't mean after you've booted up, but during the boot process and until HDMI device drivers are loaded.
I do not see an easy way out of this other than borrowing a VGA monitor (if your machine has that output) and using it until you can get the HDMI device driver issue(s) sorted. Otherwise, you're probably going to have to commit to fixing the laptop (I presume this is a laptop) monitor.
Has this arrangement made it through Feature Updates past? I ask because it would be unusual for it to fail now, though not impossible.