I had a BBC that the fuel pump lobe on the cam was flat, that why it had an electric on it when I got it, other lobes were worn too.Sounds like that may be your problem.Dan'l
If you look down the square hole the fuel pump goes in you will see this round cap on the end of the cam that is off center so when the cam turns it pushes the fuel pump shank every turn so it pumps fuel out the outlet.If that piece is worn or the shank coming off the fuel pump is worn it will not push the fuel pump shank enough to pump any fuel out the outlet of the fuel pump. Can you provide a picture of the fuel pump shank?