I can't tell from a google search what kind of interface it has, but I'm going to go ahead and say now that it WILL work.
Don't pay an LBS to do this, the internet is you friend. For the amount of money you'd pay them, you can buy the necessary tools. Also, it won't take 2 weeks
If they are both ISIS or octalink or outboard, then all you have to do is swap the cranks and can leave the BB. Since one is shimano, it's probably octalink, and since the other cranks are cheap, they're probably ISIS.
Here's what you'll need
8mm Hex
Crank extractor (I hope you can get one to work for both ISIS and octalink)
bottom bracket wrench (or one of the hook/pin things).
Remember as you sit on the bike, for the BB cups, the Right side is reverse threaded (righty loosey, lefty tighty)
while the left side is standard threads (this is the opposite of how the pedals are threaded onto the cranks).
Take the bolts out of the center of the cranks, use the crank extractor to thread into those holes, and tighten it down to pull off the cranks/arms. Then remove the BB using the tool, remembering how the cups are threaded.
Take the shimano crank over to the other bike (which you've done the same thing to) and reverse the steps. It's worth putting some grease on the BB threads, and the BB splines, it helps with creaking.