When the BIOS ignores the F11 key and starts Windows or errors, it probably means HP Recovery has lost its hotlink due to clean reinstall, reimaging, startup repair, etc.
What often works with HP is to then mark the Recov partition Active then reboot into it directly or by tapping the F11 key. If it will do this it will likely run full Factory Recovery with or without file backup. Partition - Mark as Active (Method Two) HP Backup and Recovery Manager
It is even a one-time option as if you bail out of it it will pass the Active flag back from whence it came and start Windows normally. If not just mark SysReserved (preferred) or Windows 7 partition Active again.
I remember Sean had mentioned this in general over a year ago and I'd been waiting to see it in action, and it worked perfectly on two HP's in the past month.