Do you struggle with canter transitions generally on other horses, or just Billy?
As mentioned above, it can be really helpful to look over your outside shoulder as this shifts your weight to the right spot and helps him pick up the correct lead.
When you fail to get a canter what does he do? Trot really fast? Or does he pick up the wrong lead?
Is he very balanced at the trot? Sometimes the transition is hard to make for an unbalance and unfit horse.
Does he know voice commands? Many horses know "can-ter!" as an aid to help back up the leg, others respond to a kiss. I find that very helpful especially with a horse that isn't sure as it gives them an extra hint.
Will he pick up canter for your instructor or other riders? Sometimes that will help to 'switch him on' to what you are doing, if your instructor can get a canter out of him a couple times and then you hop on, he will be more likely to do it for you and then you can gain confidence that when you apply the aids you will get the correct response.
If despite all of this you still don't get the upwards transition - if he is balanced, physically fit and capable of cantering, your aids are correct, but you still don't get it - he may be taking the mickey and ignoring you. In this instance a tap with the crop on the shoulder can help. Leroy can be like this - I cue for canter and he just continues trotting faster but a good sharp smack with the crop and he jumps forward into his canter, then each time I ask for it next there's no need for the crop. For a long time I was hesitant to do this though, I thought the failure to canter was because I hadn't cued him properly. Its hard to know sometimes.