Now both my Windows 10 machines are running the Anniversary Update and Windows Subsystem for Linux and Ubuntu for Windows.
The Storm-puter had to be encouraged a little bit, as it was stubbornly refusing to automatically download the Anniversary Update.
So I clicked on Learn More, as recommended by Microsoft here, and that got me the Windows Upgrade Assistant, which cheerfully pulled the update down for me.
From then on, it was smooth sailing, and now my giant 'tar xzf' command is happily running away, verifying a massive backup that's been waiting a month or so to be expanded onto the new machine.
Next step (though not tonight), is to follow these instructions to see if Java 8 is happier with the latest patches.
Ubuntu for Windows. It's cool. Check it out!