I ran as admistrator by changing the shortcut properties and running the the sortcut directly. This seemed to do the trick Athough annoyingly it now asks me every time if I want software from an unknown distributor to make changes to my computer.
I was running before from GOG Galaxy so I changed the shortcut for GOG Galaxy to administrator and tried running Re-Volt from there again. Bizarrely, some (all?) settings I changed from running the shortcut directly or from GOG Galaxy aren't retained if you use the other method. I don't know what is actually happening when you hit the "Play" button in GOG Galaxy; no clue if the game is being run as adminstraor from there. And there's one setting, Non-linear steering for my XBox controller, that isn't sticky no matter how I run things.
Anyway, things seem to be working well enough now that the game should be playable. It might take me a while to beat a challenge time or win a race to see if the progress table is being saved though :-) This game is fun, but hard.
Thanks very much!