Once you're west of Newbury, the fastest, flattest and simplest route is to follow the A4 (which is pretty quiet at most times of day in this section) as far as Chippenham, then the A420 into Bristol. But you probably could have worked that out for yourself! And while not bad (certainly a lot nicer than it sounds), there are far nicer ways.
So, again starting from Newbury (can't you help you between there and London, sorry) go through Kintbury, Hungerford, cross the Thames in Hungerford and take the B4192 towards Chilton Foliat. After something like a mile and a half on a right hand bend with a house on the inside corner, turn left – visually straight on – to a minor road. A mile later where the road bends left you turn right into the grounds of Littlecote House (country hotel) (it's a public right of way and I've never had any trouble riding through even though it looks private). Follow the drive through the grounds and where it turns sharp left you go straight on following a track, quite rideable. This brings you onto a road at the bottom of a steep hill to your left: you turn right into Ramsbury. L at T, follow the road through Mildenhall (not that one!) into Marlborough. Coming into the town go straight on at the first X-roads then left at a T. This brings you into the town centre / market place. Turn right and follow the A4 for a few miles.
In West Kennet, before you pass the long barrow or Silbury Hill, turn right onto the road that leads along an avenue of standing stones into Avebury (expect many hippies at this time of year!). In Avebury take a left by the clump of trees just before the Red Lion, follow the road and some tracks (a bit bumpy but perfectly rideable on a road bike) through Yatesbury into Cherhill. Follow the lane out the back of Cherhill and then a bridleway past some gravel pits into Calne. From Calne there's a cycle route 403 on a disused railway line, pretty decent, taking you into Chippenham. Then through Biddestone and Colerne, down Bannerdown into Bath and Bristol-Bath Railway Path.