Iain: Seeing as there have been no guesses so far a clue is in order.
The company is around still today and releases things on tablets etc. They currently have a kickstarter campaign and they started out on the ZX81 and then onto the C64 and Spectrum.
Ghorpm: If nobody answers within several hours I will :) I don't even need the hint - I know quite a lot C64 games. There are, however, two genres where my knowledge is limited: SHMUPS and racing games but your game is definitely not one of them...
Yeah, you are like me with the C64 stuff, I recognise a lot of it myself with still owning and using my C64 on a regular basis.
Give it another day to see if anyone gets it then let everyone know!