It's likely that the pump's operating contact is not opening.
Operation is as follows:
1- Pump energized
2- Pump solenoid coil is energized through the top electrical contact
3- Pump solenoid pulls the diaphram up against spring pressure
4- Fuel is pulled in to the pump chamber via one way disc valves
5- Pump solenoid & diaphram reach the top of their stroke
6- At the top of the stroke the electrical contact toggles (snaps) open & de-energizes the solenoid
7- The pump spring pushes the solenoid & diaphram back down & fuel is ejected via the one way valves
8- The top contact snaps closed & the solenoid is energized again for the next stroke as in #2 above
Frequently the diaphram loses elasticity & prevents the solenoid/contact from traveling far enough to open the top contact.
The diaphram is positioned on a threaded rod so that it can be adjusted for correct travel/contact operation. There is a specific procedure to set the diaphram in relation to the top contact to assure correct contact toggling.
Your chances of getting the old pump operating reliably without using the correct rebuild kit are slim. The rebuild kit should have instructions on the set up & adjustment.
I'd really suggest that you have an experienced pump rebuilder do the whole job. One of this forum members, Dave Dubois in Bremerton,WA I believe, can do the job with a much better chance of long time reliability. It's likely that British Car Specialists in Stockton can supply the parts & rebuild your pump also.
Look here:
https://homepages.donobi.net/sufuelpumps/
D