The protector of a battle with the Siege subtype is not that battle's controller.

If damage dealt is being divided or assigned among multiple permanents or players, that damage is divided or assigned before any effects that modify how much damage will be dealt. For example, if you attack the chosen player with a 5/5 creature with trample and it's blocked by a 2/2 creature they control, you can assign 2 damage to the blocker and 3 damage to the defending player. Those amounts are then doubled to 4 and 6, respectively.

If another effect (or effects) modifies how much damage a source would deal to the chosen player or a permanent they control (for example, by preventing some of that damage), the player being dealt damage or the controller of the permanent being dealt damage chooses the order in which any such effects, including the effect of Sawhorn Nemesis, apply. If all of the damage is prevented before Sawhorn Nemesis's effect would apply, its effect no longer applies.

The damage is dealt by the same source as the original source of damage. The doubled damage isn't dealt by Sawhorn Nemesis unless it was the original source of damage.