I'm investigating what methods peoples of the past used to travel upstream before Steam Power become available. More specifically, I'm wondering how the people who navigated the Mississippi and the Nile rivers would have done so, particularly with vessels of sizes larger than personal craft (small to large scale trading ships). From what I can discern, these two have their own difficulties that make answers for other rivers seems less likely. Namely, the Mississippi is too large in many places for beasts of burden to pull the craft along from the shore, and the Nile shares this problem with the Mississippi in addition to how it runs counter to all prevailing winds, meaning that sails would be all but useless.
In the case of the Nile, your information about the prevailing winds is incorrect. The current of the Nile runs from south to north. The prevailing winds along the Nile blow from north to south, although this is more consistent south of Cairo than it is in the Nile Delta between Cairo and the Med. In the Delta, the prevailing winds from the north are most consistent in the summer. In the winter, the problem isn’t so much acountervailing wind, but actually that the wind might not be consistently strong enough to overcome the current.
The seasonal component was also relevant in the pre-modern period because of the annual Nile flood, which significantly increased the speed of the current and the height of the water.
In the absence of wind, or a northern wind too strong for the meandering current of the Nile, Nile boats could be towed, rowed, or punted (I.e. pushed along with a pole.) The predominant form of towing wasn’t with “beasts of burden”, it was human labor.
It’s also perhaps worth noting that in the Delta there were/are a variety of artificial canals that would have made this easier under certain conditions (I.e. when they were navigable during the annual flood).
Sourcewise, see The Medieval Nile by John Cooper