i know from experience that a regular 1/2" pitch chain will NOT work with a skip tooth (1" pitch) sprocket. two totally different animals. the skip tooth chain has two different links, alternating 1/2 link, then 1' link, repeating... a newer 1/2'' pitch chain will roll right off, jam up, or break.
i've been down this road before and i've tried just about everything imaginable.
here's some options for it that you're just gonna love...
the easiest thing to do, would be to buy a "newer" sweetheart sprocket with the conventional 1/2' pitch. you can get one here for 6.95.
Pedals ~ Sprockets ~ Cranks
that way you can use a new chain, the new sprocket on your wheels, and everything would be nice and modern, and simple.
but who wants to take the easy way out?
so, the first thing you have to do is find a skip tooth, inch pitch chain. if there's one rusted to the bike already, you're in luck. soak it in oil for awhile.
if not, ebay always has them. they usually go for 50 bucks for an old stretched out one, to 100+ for NOS.
that's the easy part. it just takes money.
now, for the rear cog (sprocket).
i've done both of these on a bendix coaster brake, and they both worked.
the bendix cog has three nubs that slide into grooves in the hub, and then are held in by a snap ring.
you can find an old skiptooth cog with the same inner diameter to fit over the hub. find one with just a round hole, no nubs.
then, weld on three nubs to match the hub. install, and ride around.
this one's even more fun.
buy an extra rear cog to fit your bike, make sure it has an even number of teeth, then grind every other one off. make sure you quench it in water after each tooth, or whenever it gets really hot, so it doesn't warp.
the only problem with this method (besides the mess, the time, the P.I.T.A. factor,) is if your using an old chain, the new cog teeth might not get along with it, so you might have to do some extra grinding and filing of the teeth. but, inch pitch chains are remarkaby strong, and don't stretch anywhere near the amount as the newer 1/2'' ones do.
here's things that WON'T work.
-switching the guts from another hub into your new one (unless you find the "holy grail" hub that i never could.)
-inch pitch chain on 1/2" pitch sprocket, and vice versa.
-finding an old rear cog that matches the one on your new back wheel, and fits right on (another "philosopher's stone".)
i don't have the bikes i did this to, so sorry, i don't have pics.
but i do have a brand new 18 tooth (equal to a 36) WALD skiptooth front sprocket, inch pitch chain, and an 18 tooth bendix rear cog, which will soon be grinded down to a 9...
hope this helped.