Surprisingly, what I've used recently to refurb the base of a few vacuums I got my hands on was headlight polish paste. My mum got her hands on some somehow, it's a bit like Cif, but instead of scrubbing and scratching as such it's meant to polish. It's got some potent chemicals in it too, but it genuinely makes plastic look brand new. Have a look around, maybe ask in some local car parts store, car wash, etc?
I don't have any pics to prove but I was genuinely amazed at how well it worked removing surface paint as well as deep scratches leaving the surface smooth and shiny.