Hydro-lock seems unlikely, if it was only on its side for a few minutes, unless it was some old engine that was already burning oil thru leaky rings or stem seals. But still always best to check all fluids, then remove glow plugs or spark plugs from all cylinders prior to cranking, to ensure all cylinders are dry enough to run.
If the smoke had lasted several seconds after re-starting, I'd have guessed a small amount managed to leak into the cylinders while the machine was on its side. After all, it's not like piston rings and valve stem seals suddenly become porous, just because the machine is on its side.
Someone mentioned coolant in the cylinders. Tell me how that could happen? I'm having trouble understanding how there's any path between cylinders and coolant, no matter which way you flip the thing. The coolant on these machines is a closed and pressurized system like any car, right? If there were some leak path dependent on gravity, your system would never pressurize, and the coolant would boil off.
Oil on exhaust seems possible, esp. if somewhere downstream that's not super hot, as that's the only way it'd keep smoking after several minutes. Still, seems unlikely, unless there was a lot of it.
Damage due to hydro-lock, although unlikely in my opinion, could cause continued smoking and the stalling under load that the OP describes. A bent connecting rod or broken valve stem would be sources of continuous oil intrusion thru piston rings or valve stem seal.
I work on gassers, not much experience with diesels. Can you run compression and leak-down tests, to see if anything serious was damaged? Short of that, if you can't find any obvious external source for the smoke (eg. oil on exhaust), then I'd be pulling the valve covers to check for damaged valve stems, etc. Last resort is pulling a head to check pistons, but at least on a gasser, you can usually debug that by compression and leak-down tests.
