How Can I Make Sure My Home Server Stays Up As Long As There's Power?

You don't need a timer, BIOS should have some kind of 'auto-boot after power cut' option. I've no idea if this still exists on modern EFI boot structures, but all my PCs are set to this in BIOS…

Inside the BIOS menu, look under the “Advanced” or “ACPI” or “Power Management Setup” menus for a setting named “Restore on AC/Power Loss” or “AC Power Recovery” or “After Power Loss.”

This is different per manufacturer, so you may have to hunt around for it.

To complement this, you will need the Windows user to auto-login to the required account, otherwise it will sit at the login screen until you do it manually.

Quote from TechNewsWorld - How to Set Up Your Computer to Auto-Restart After a Power Outage