I am medically qualified- had a mate who is also a Dr who had the same issue after being set up by a bike shop in terms of his riding position, when he took up cycling- symptoms were exactly the same, developed on bike, ached for ages after, lateral (outside) aspect of both knees, and took at least a week to settle down each time.
I put his seat up, and moved saddle position slightly, meaning his legs went up and down in a straight line and not pushing outwards at the top of the pedal stroke, which essentially caused his ankles to supinate in compensation and stretch the lateral ligaments of his knee. Without looking at your position I cant diagnose or say for sure its the same issue but trying it is certainly free of charge!!