Losing an hour's work from hardware failure isn't too bad in the grand scheme of things.
Browser backups are stored internally to the browser, so the only way to recover it is to run the same browser that you made the backup with. If you've moved to another system then I guess you don't have access to that. If there is a way to solve it, I'd guess it'd take longer than an hour, so you may as well just re-do the hour's work that was lost.