I think I've found it. There's a turning off the A403 before the roundabout which leads to an acess road that seems to go upto the toll booths.
That'll take you to the path that's on the southern side of the bridge. Both times that I have used this bridge, I've taken the path that runs on the northern side of the bridge, as described by mickle.
what mickle said. and watch out for errant mopeds and motorbikes.