The usual cause of a squeak from the front (nearside) on an auto is the subframe mount tears out of the chassis rail underneath the airbox.
I don't see how gearchanges would affect it though, unless you are really making it go and it is the driveshaft straightening out moving the subframe, as the engine/gearbox are secured to the chassis, not the subframe.
My mechanic told me 2 years ago that my subframe mounts needed replacing, but since then, the car has passed 2 MOT's. Maybe they really do need replacing now then. Even when I accelerate gently the squeek is there.