I walk for 15 - 20 mins before doing anything more
It takes 20mins for all the muscles to properly warm up before you should start jumping or doing dressage movement.
It depends on the horse. My Arabian I walk for 5-10 minutes, then trot for 5-10 and canter on a long rein, then I go back to walk and pick up the reins for exercising. The chestnut is very hot, so I walk a few minutes then pick up my reins and start trotting and cantering quite quickly. She has a tendency to spook and buck when she gets bored. With her I only do short amounts of walking, and a lot of transitions to keep her occupied 😊
I think there is not a limited time for the warm-up, It depends on the horse, his age and his physical condition !
Also during the warm-up I'm training to concentrate my horse and relaxing him.
In my opinion, the first part of your training session is the most important part of the work! In fact it's a preparatory phase for next exercises ! Usually i walk 20 minutes, then 10 minutes of trot and after a little session of canter! I have to feel my horse ready for working :)