Inertia switch, relay pack, Fuse, broken wire, or just general bad connections that the Mini always suffers with.
The fuel pump fuse is tucked up under the scuttle panel somewhere, its an inline jobbie.
The inertia switch is the black thing mounted on the bulkhead, press the top down, if it clicks, it had tripped. Check its continuity with an ohm meter.
The relay pack is difficult to check, but if you are VERY careful, you can remove the cover, and not letting it touch any of the bodywork, get some one to watch the relays as you key the ignition on, one relay should energise for a couple of seconds and then de energise, if it doesnt, the relay pack could be faulty. If it does, but the pump still does not run, you need to check the voltage at the inertia switch, if there is no voltage there, you need to check the voltage at the fuse, no voltage there, you need to check the voltage on all the brown only wires at the large connector on the relay pack, no voltage there and the fuse link may have blown. Remember, that the pump only runs for a couple of seconds when the ignition key is turned and while cranking, if the engine is not running, the fuel pump wont be.
Broken wiring especialy on the fuel pump circuit is common.
Other areas to look are the large thirteen way connector on the bukhead, the thirteen way connector under the dash on the passenger side. Check for no bunt out contacts. The loom runs from there to the rear of the car under the carpet along the floor on the passenger side, into the rear passenger side bin, behind the rear quater trim and into the boot behind the tank.
The Mini Fusebox is always a problem area. Take it off and have a look at the connections on the back of it