( Just googled the pump. The one that came up looked as if it is driven by a toothed belt - probably the cam belt, but might be a separate belt. )
As I said previously, I'm not familiar with this engine so I was suggesting general faults which might give this problem - although I just spotted that you say you've checked the pickup strainer, so sorry for suggesting that one. (Does that mean you had the sump off? If so, presumably you didn't spot anything nasty inside like bearing metal or other s**t.) Does the engine run, or are you spinning it up on the starter? Not good, as I'm sure you'll know, to actually run it if you think there's no oil supply!
Having Googled that oil pump I'm really intrigued by it. Looks like it is mounted on the outside of the block and driven by the timing belt? Just as you'd expect an older type injection pump to be mounted. Only here we have the oil pump first with the injection pump piggy backed onto it and a shaft right through to drive it? My "pugglt" old brain can't remember having seen this setup before. Quite like the idea though as it means the pump must be turning if the engine runs, unlike some of the VAG stuff I've worked on where they use a separate chain drive off the front of the crank to the pump underneath. The chain can break and you're non the wiser but engine goes on running - for a short while anyway, till it all goes bang! (Or screech as it siezed!).
Here's a wee edit addition. Just remembered. Around the 60/70's Vauxhall had a slant 4 cylinder engine which had its oil pump high up in the block under the distributor. Mostly found in the Victor and performance orientated versions of the Viva and Chevette, CF van too if I remember. I think there was even a Lotus development of the engine. Being a Leyland (BMC) dealership we only occasionally saw these vehicles (often a trade in) but I remember well that if you fitted a new oil pump it had to be well primed with oil before fitting or it wouldn't suck the oil up from the sump. It was very important also not to leave the old oil draining too long at a service otherwise, with the pickup pipe uncovered, oil would drain from the pipe and the engine would run for some time on restarting before it could bring up some oil pressure. You could hear the ends "knocking" away gently until the pressure suddenly quietened them. I quite liked these cars to drive but never considered buying one for this reason. Always fancied a Ventora though!
So I'm just sitting here thinking "that Fiat oil pump is quite high up on the side of the block" wonder if there is any tendency for these pumps to airlock the pickup side? Anyone?