I just read your situation and I also replaced the oil pick up tube with new gasket after doing the timing chain, and I did the following with no problem and I also got oil pressure showing on the oil pressure gauge:
First and most important disconnect the orange wire behind the battery, and you must have an oil pressure gauge in order to make sure you do built up oil pressure before you reconnect the orange wire for start up.
I didn't pack the oil pump with Vaseline as most do
I first filled the oil filter as much as I can and then put 3 quarts oil in oil pan , then I remove the lower oil cooler from the radiator taped it against some pipe so the oil cooler is vertical position. Then place a small funnel inside the oil cooler hose, you will need a Chevy oil pump prime tool and place it inside the Cam Sensor hole; make sure the tool slot is fitted on the oil pump shaft. DO NOT start the engine, if your oil pressure gauge is electric then just turn the key to the "ON POSITION" , and have someone inside the car telling you that you have oil pressure. Attach the cordless drill machine to prime tool and run it at CCW position, at the same time fill the oil cooler hose with about 1-1/2-2 quarts of motor oil, or until you have oil pressure present. Once you have oil pressure, re-install the oil cooler hose back onto the radiator; set the cordless drill machine to CW until the other person inside the car tells you that you oil pressure on the gauge. Remove prime tool rotate crank shaft to new marking of 25* after TDC install the Cam Sensor as per Casper's Instruction.