This is the time to pull out some big math, or just make up stuff as I go along... Let's postulate a flow of 30,000 cfs out of the dam, then pick the narrows between Warm Creek and Gunsight Bays. The lake is about 1,000' wide there, and about (very roughly) 300 feet deep. So the cross section is 300,000 square feet, so 30,000 cfs divided by 300,000 square feet gives 0.1 feet per second, or 0.068 mph. I doubt that you would be able to even measure a current that small.
The nominal "residence time" for water in Lake Powell is 7.2 years, so the water takes about 7 years to flow from one end of the lake to the other.
The current will be proportionally faster where the water is not as deep, notably up above The Horn. There has been some post-filling earth movement near the Rincon that left a shallower sill, and there might be a noticeable current there at minimum reserve pool and full spring flood flows.
It's nothing like the St. Lawrence River with currents in the high single digits in the navigation channel, though.