The SW is correct original equipment for many 3-speed bikes of a very specific vintage in the mid to late 1950s (1956-58?). The hub shell of the SW is smaller than the AW. However, the difference is small enough that you play with the wheel a bit (interlacing, using rim/nipple washers, etc.) you can re-use spokes. The stock spokes going from SW to AW will be slightly too long, but if you're creative building the wheel a little bit, you can get it to match back up. If you've already bought the parts, might as well try to get the SW working properly. You'll probably need to tear down the hub and clean it out.
The SW relies on crescent-shaped, floating pawls to engage the transmission core to the hub shell (the AW uses finger-shaped pawls and tiny, wire springs). Any kind of grease, dirt, sticky crap, etc. will prevent the hub from working properly. Any substantial wear on the pawls will keep the hub from working properly as well. But now that you've bought the parts and they're on the way, I'd try to get it working. It's original to the bike, and at this point it's just some time and effort to see if you can get it to work well. Get the parts together, give the hub a thorough cleaning and oil with very light machine oil (no medium/heavy oil or grease on the pawls).
I have a very lightly used 1958 Raleigh Sports that was in great shape, mainly because the SW hub stopped working and the bike sat after that. I gave the SW a try and could not get the pawls to engage properly even with a proper cleaning and set up. I switched to an FW hub (same hub shell as the AW). But every so often someone shows up with a nice, working SW. They're really quiet and smooth-running when everything is good on them. But there's always that chance you get one that slips out of gear. It's murder if you go to really lean into the bike and those SW pawls slip on you...