Some computers won't start with a dead CMOS battery, and others will. I had an old Compaq Armada laptop with a dead battery, it would start but required you to change the date and time every time you turned it on.
You should probably buy a new battery and see. They are so cheap, only £1 for about 5! You need to get a CR2032 battery. Alternatively, just do a search on eBay for 'CMOS battery'.