I also greased and reassembled the rear hub, and now have a functional rear wheel that's in much better shape than when I found it. Here's the before picture for a comparison.
I've even tested the coaster brake by spinning the wheel in my hands, and it seems to work just fine. Like I said before, it's not quite as shiny as I would have liked, but there's only so much one can do after years of weather.
On a side note, after exchanging emails with a very nice gent and former mechanic in England (who remembered when Runwell was still around in the 1960s as a smallish parts distributor), I believe I'm dealing with what's left of the original nickel plating, rather than chrome, which does a little more to explain why it was so badly deteriorated. Chrome was apparently introduced in 1928, but nickel plating was still applied after that date, so it remains hard to pin down exactly when my bike was manufactured.