If you look at this diagram,
http://www.crossfire.homeip.net/twinnie/Image3.jpg
you will see that the output of the Oil Pressure Switch is tied directly to the fuel pump. Another output goes to the Oil Pressure Gauge. If the oil pressure switch cuts off the output it will not disable the fuel pump, but it will send a signal to the ECM that there is no feedback from the fuel pump relay.
The result is that the engine will start with the ignition key on, but after starting if the signal from the oil pressure switch isn't there if will shut off the fuel pump.
These two outputs (the switch, and the gauge) are not tied together. One sends an analog voltage to the oil gauge, and the other is simply a on-off contact switch to tell the ECM that there is oil pressure. So if one is doing something weird, you really don't know what the other is doing without putting a meter on it.
You can print out the diagrams and tape them together to get a whole system layout.
http://www.crossfire.homeip.net/twinnie/system.html
All these sensors have a job to do, and that is to report conditions back to the ECM. If they aren't working properly, or the circuits are broken, the ECM will not know what to do.
One other important thing to note is that the oil pressure switch is reference grounded to the engine block. If the threads into the block have any kind of sealant, or teflon tape on them, it will keep the switch from having a proper ground and will result in the switch from working properly.
So what you can do is put a meter on the Brown/white wire on the oil pressure switch to the fuel pump relay, and make sure it has +12V on it. Then go to the fuel pump relay pin E and see if the +12V is there when you start the engine. If not, then the switch is bad. If there is voltage on pin E, but no output on pin A, then check your fuel pump fuse. Pin A comes off the Fuel pump fuse.
Then check Pin B on the relay. This comes from the ECM to tell the pump to keep running, and pin C is the return signal to the ECM. If you put a meter between B and C there should be +12V when the engine starts, and +12V on A at the same time.
If there is +12V on between B and C, but no voltage on A, then your fuel pump relay is not making contact.