It really depends on how he was trained for WP, if that was all he ever did, or if he knows the diffence in seat ques, etc. I have 2 examples. My 3yr old Morgan who is a western performance horse, and my 16yr old Arab who is a mix of everything.
I trained Storm (my 3yr old) to have a nice western gait, I don't beleive in the traditional lope that is stilted so ask for more forward motion. He does a beautiful walk, jog, lope. Now that being said, when I post he knows that I want more forward motion at the trot so he will give it to me. If I ask him to frame up more like an english horse then he gives me more forward motion in the lope. For him the transition to english jumping will be very easy because he knows the difference in ques that I have already put in.
Now take my 16yr old Arab Mercadies. We have only had him for about 4yrs but we know pretty much what he has done based on how he responds to different situations. He does (surprisingly) very well in a WP class as he will jog and he can lope almost on the spot it's so slow. Now I'm sure he was never tought to have a western head set as he ALWAYS carrys his head high but that is also part of his bread and I have no need for him to learn it now. Now he also has amazing forward motion, he loves to game, go fast, and make hard turns. He can go from that slow WP lope to a full out, charging down to the barrels in seconds, and then he can go right back down to the WP lope. For him it's ALL in the ques and how you set him up. He also jumps, I believe he has don't both Hunter and jumper just based on how he goes. We jumped him a little when we first got him but he was abused before we got him so gave him time off of that. This past summer I decided to give it a try again, we did a 2'3" course and he knew what he was being asked no problem. Again if I change how I ask for things with him then he will go to whatever it is that I want.
So I guess the point of all of this is to say that if you have slightly different ques for the different things you want to do (ie: how you hold your reins, how you sit, etc) then the horse will pick up on that and know what you are asking at any given point. You should have no problem going from one area of riding to another.
Hope that helps, and sorry that it was so long :)