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Found a nice workaround: JoyToKey.  It allows game controllers to emulate keyboard input (or mouse), so it was easy to assign the keys Blastaway uses to my joysticks. It recognised all my 3 USB devices. Well worth the $7 asking price.

Alternatives: Xpadder (couldn't find a test version so I didn't try it) and InputMapper (seemed a bit complicated so I didn't try it).

So... I can now finally start playing Blastaway!

Great to hear you've found a solution! Thanks for having shared it!

Have fun!