You may possibly only have a 'valet' key.
My locksmith explained to me about valet keys (as in U.S. valet parking).
They key is the same number as a proper key, but cut on a different blank, which is (I am guessing) maybe shorter, and only locks and unlocks the drivers door and starts the car. The idea is it prevent a valet parker stealing stuff out of your boot (trunk!) or locked glovebox.
Does your key open the boot and glovebox? If so it is a not a valet key and you may have a central locking fault.
If it does not open the boot and glovebox, it is could be a valet key and the next step might be to get a proper (non valet) key cut to the same number.
If you need to do this you do need a proper locksmith with access to the correct Mazda key blank and a computer controlled key cutter that can cut from the key number. I don't think multi-service places like Timpsons can necessarily do this (the half dozen I tried on my quest for a key last year couldn't)
There is a place here in West Yorks that can do it:
http://www.harfordssecurity.co.uk/
Although they will do mail order, you can no doubt find a security/locksmith firm locally. You can find a master locksmith near you on the Master Locksmiths Association website:
http://www.locksmiths.co.uk/
There is a factsheet showing pictures of the keyblank types in the members area.