We have partial progression.
Feeling somewhat safer by the post from frostbite - I plugged the dongle direct into the back of the PC (actually, I first tried into a small USB hub - but got a ratty message saying insufficient power).
PC found the dongle immediately. I started the bluetooth "add devices" and whilst it did find the GPS unit over on the filing cabinet (which I'd forgotten about (that's the GPS I forgot, not the filing cabinet)) it did NOT find the mouse.
Off to the cupboard for a new pair of AA's and the bright red light came on under the mouse.
Back to the "add device" software - did another scan. Found the PDA (which I switched on purely for spite - but will continue to use through its cradle (unless you advise otherwise)) and found the mouse.
Followed all the trails for mouse installation - software suggested all was hunky dory - offered me a passcode (declined) then I clicked "OK". Software instantly 'forgot' the device. Not there anymore. Played the game again - same result.
I do still have the original mouse plugged in - a trackball on a USB plugged through a PS2 adaptor (at least until I solve the mystery).
The new mouse is a Microsoft 4 button plus tilt/turn scroll wheel. - it came as a kit with the dongle and worked fine on the laptop.
Humphhh!