Just some input - it sounds very like in the style of 'Georgie Fame and the Blue Flames' but, just different enough from one of his records (can't remember the name) and, probably a cover group/version?
Well, it sounds alot like Yakety Sax to me which was performed by Boots Randolf. You probably remember it most from the Benny Hill show, but it was recently used in V for Vendetta. That's probably the first place you want to look (downloads from Limewire anyone?)
The song seems to be a copy of 'Get Away' by Georgie Fame & the Blue Flames, with a similar tempo but different lyrics, instrumentation and an unknown vocalist - it certainly is NOT Georgie Fame. The saxaphone portion sounds vaguely like 'Yackety Sax' - but again isn't. Maybe the composer needed to have some dance music for this scene, & created this curious hybrid after failing to produce a credible sounding 'pop' version of the recurring main theme.
The soundtrack to 'Grand Prix' isn't currently available, but this song wasn't on it. Maybe it will be reissued if the sales of the DVD are healthy.
Unless you want to contact the studio or the composer Maurice Jarre, It seems the singer and performers of this tune have been lost in the mists of time.
On the other hand you could buy a compilation CD by Georgie Fame & the Blue Flames with 'Get Away' on it.