I followed what you did, and that worked.
Literally in seconds the car alarm started going off, allowing me to open the car doors.
I purchased a battery from Euro parts online, and the terminals were in the wrong position. I managed to buy another one at Halfords tonight.
Tomorrow morning, I'm going to attempt to place the lightbulb back. (I destroyed the plastic mould holding the wires)
Thanks Clifton94 and Tigerfish! :)