Hi Sylvia,
Many thanks for the pointers. I didn't get a response from the e-mail address (at least not yet) but some of the other links led me to the solution. I need to place a bootable UEFI image in \EFI\boot\bootx64.efi - this file doesn't actually do anything, but the UEFI firmware detects its presence and then shows the NVRAM entries added by efibootmgr.
I did this in the UEFI shell by running this command:
mkdir fs0:\EFI\boot cp fs0:\EFI\grub\grubx64.efi fs0:\EFI\boot\bootx64.efi
Now my machine boots into GRUB and then Linux automatically!
I see what people say though when they talk about the current crop of UEFI firmware being quirky... This and the number of bugs I am finding in VisualBIOS lead me to think that there hasn't been much testing done at all on any of this!