Depends on how they are being taught. My partner has just started to learn on my mare and is a complete beginner. He is having a half hour private lesson every other week and practices about twice a week. He's been riding now for about a month and has just got the hang of rising trot (in balance) and has started to learn how to move around the school. I would say give him another couple of months and he'll be competant in walk, trot and canter.
Ride and be compentant in walk and trot usually takes about 4 lessons in my experience. It is the step to canter that takes a while and very much depends on fitness, confidence and natural balance. But I think it would take a minimum of 10 to 15 lessons to be able to start thinking about 'controling' a horse in canter and not just sitting up there and praying
My BF is learning to ride, he mastered the rising trot after 2 lessons and was 'confident' on his own on a safe horse after 4 lessons. He has cantered a couple of times (once on the lunge, once 'by accident') and is very very wobbly. I think the canter will take him a while.
Being able to sit on a horse and steer in those paces, a couple of months. Being balanced and in control at all times, with an independent seat and hands, an awful lot longer.