A western trained horse SHOULD be able to do anything just as good as an englished trained one [and vise versa]. Sure, an english saddle might be a tad bit more comfortable to jump in, but if you don't mind the horn than anything is possible. In western trail classes students usually have to jump over obstacles.
I use natural horsemanship methods, which is amazing with jumping. I've never ridden a jump, but my boy will jump over them pretty great. (I am not comfortable with riding him yet, I have ridden plenty of other horses, though.) The training pretty much is "you will go over whatever obstacle that is in front of you because you trust me to know you are safe" sort of thing. Whether it is poles on the ground, or barrels, if I point my horse in that direction I'd expect him not to give a second thought to it unless their is any immidiate danger. (Like do not jump off a cliff, for example.)
When I start working with jumping I would lunge the horse with me on the ground. Then start encorperating jumps into it. First small poles, then maybe tubes, then maybe sideways tipped barrels, and then actual jumps or even larger logs. For movies and books just go on the internet and watch videos on youtube. Look up natural horsemanship jumping because it will give you an idea of what I'm talking about.
With the saddle, you can start introducing it during lessons. First day just let him sniff it, set it on his back and take it off again. The next day actually sinch it up, walk him around (with you on the ground) and take it off. The third lesson should be a good chance to start working with the new saddle. Do the lunging (with you on the ground) and have him trot around, get the feel. Maybe even try some jumps or poles. Fourth day, if he's taking to it well, lay on him a bit with that saddle on. Maybe get up on him and ride him around a bit. Nothing too vigorous. You can just keep on switching to the english saddle from there.
I hope things go well! I am no proffessional jumper, seeing as I have never even ridden a jump. But I'm pretty good with teaching things from the ground up. I'm sure if I hopped on my horse and tried to jump him I'd fall off by my own error. XD