Do you have a multimeter ? You can buy one quite cheaply in B&Q or wherever .
The first thing to check is by metering for voltage , this means grounding the earth wire of your meter to a bare metal part on the bodyshell (if you lift the carpet in the footwell see if there is a metal screw going into metal somewhere - this should do) , you will then be using the positive probe to look for voltages . Set the meter to d.c. volts with a range greater than 15V.
I'm not sure whether this circuit switches on supply or ground side of the lamp . The way to check is , with your meter set to volts and the glovebox switch button pushed in (off) check the voltage on either side of the lamp and either side of the switch .
If you read a constant 12-14 V (or thereabouts) on both sides of the lamp and one side of the switch , then the switch is breaking the ground return (earth switching) . If only one side of the switch shows a positive voltage and everything else shows 0V , then it is positive switching .
If you cannot get a positive voltage anywhere on either side of the lamp or either side of the switch , then the fault lies elsewhere and you would need to trace the wiring back to the fusebox .
I would expect it to be earth switching , in auto electrics , ground wires are usually brown so this will help to confirm .
Assuming you have confirmed the presence of a positive supply to one side of the circuit , we now need to check the earth continuity on the other side of the circuit . (Just to be safe , first remove the fuse that protects this circuit in case you accidentally short the positive side to ground during this check .) Now set your meter to continuity or resistance , if it has a 'beep' setting , use this . With the meter still grounded to your earth point on the floor or wherever , now test the opposite side of the lamp/switch where you did NOT find a positive voltage earlier - either your beeper should 'beep' or your resistance meter should indicate zero ohms ; if it does not then you have an earth continuity problem and will need to trace the ground side of the lamp/switch assembly back to the bodyshell .
While you have your meter on resistance/continuity , check across the lamp : there should be some continuity / lowish resistance measurable ; although it is a new lamp , it just could be faulty . Also meter across the switch for continuity pressing/releasing the switch button , looking for a change in readings ; if it does not change - stays open circuit then the switch is faulty .
Hopefully this will allow you to isolate the fault .