When did the ancient Egyptians discover or reach the Nile Headwaters?
Herodotus reports that the priests of Sais believed the Nile to draw its source from eternal springs within two great mountains, somewhere between Thebes and Elephantine, the first cataract of the Nile. On the other hand, Egyptian temples existed further upstream of this, so the ancient Egyptians must have gotten closer than that to the source.
Claudius Ptolemy writes that a Greek merchant by the name of Diogenes, setting off from Rhapta, discovered the Mountains of the Moon, whose snow melts into great lakes that then gave the Nile its source. This might have been the Rwenzori Mountains and Lake Victoria.
If Ptolemy's account is to be believed, than it is plausible that the Egyptians might have taken Diogenes' route and reached the general vicinity of the source. On the other hand, if they ever did, no record of it survives AFAIK.
Generally speaking though adventurers of Classical Antiquity did not seem to have had so much success exploring southward on the Nile.