Down the street this afternoon to pick up oil for the BSA, and a spark plug for good measure. I settled on 15w50, though apparently they're not picky. Per instructions from some helpful forum members, I verified the oil pump was working after draining a whopping 200ml or so of oil from the sump, with the upper tank being bone dry. It looks like a leaky sump plug was responsible for all the oil on the bottom of the motor... what'd Mater say about old British engines? "If there ain't no oil under 'em, there ain't no oil in 'em."
After sorting out the oil situation, I connected a far-too-tiny 6V battery, pulled the plug and replaced it (Champion L78, NGK Part #B7HS) and held it to ground to check for spark... which I didn't find. Checking the coil, it looks like the primary side might be short circuited. No lights or anything either, but that's not really that surprising.
It's a bit warm to be mucking with the thing, and we've got Christmas celebrations coming up so it'll probably have to wait until the new year to get anything done.