Wow! a 3800 into an MGB. That thing is gonna go like a rocket when you get it running. (BTW, the MGB is a British-built, Morris Garages originally, two-seat sports car with god-awful electric by Lucas. Built through the sixties and maybe part of the seventies) [edit: Do you think the rear-end can handle the extra torque? The original engine put out, what...100 ft-lb?]
Anyway, the oil pump in the 3800 is a gerotor driven directly from the crankshaft. As far as I know, there should be nothing electronic preventing the oil pressure from building. If the pickup pipe to the sump is not tightly fixed, I suppose that would let it suck air, causing the problem. Or, if the relief valve assembly was removed, or stuck wide open?
[edit 2: The service manual says when you've had 'em apart, you should lubricate the gears and pack the gear cavity with petroleum jelly. I'm guessing this is to prime the pump on a dry, rebuilt engine. I've never had one hang up as you describe, but maybe it was sitting for a long long time?]