just a guess that new owner is trying to get oil pressure but spinning the starter did nothing, zero nada.  Is there a thread on priming the oil pump on here that we could point him to?  Basically its pull the bypass, make a little paper funnel and tape it to the block,  pour in some oil, turn the crank backwards, repeat a few times.  Unfortunately, could already have damaged many parts that dont like to spin without oil pressure.  Oh, almost forgot, getting a dry bypass out can be a pain, I had to make a tool.  Note to self...where is it?

likely a pin drive for a pre A+ motor, but you won't know for sure until you remove the old one. Who knows if it is the original engine and the original cam. I guess you could buy one of each...

Pin drive or slot drive.

https://www.minimania.com/part/GLP139/Austin-Mini-Oil-Pump-Slot-Drive-For-Small-Bore-Motors

https://www.minimania.com/part/12G793/Classic-Austin-Mini-Oil-Pump-Pin-Drive-Small-Bore-Motor


edit: I guess I should add that I doubt you will find that just replacing your oil pump will solve whatever problem you are trying to diagnose. Once you are at the point of replacing your oil pump (ending out and flywheel housing cover off) you may as well split the trans off and pull some bearing caps. There's some things you could check even before you go to replace the oil pump, but I guess that would take more effort on your part to ask what those might be.

Purchased a 1976 Mini and I believe the oil pump is dead! Its a 998cc...anyone have a part number to save me some time? TIA!!!