Knowing nothing about your particular interests, this is just a shot in the dark. Why are your anchor points Bath and Edinburgh?
Assuming those cities & dates are set in stone, you're not giving yourself a chance to do much sightseeing on Days 5-6. You basically have to go into London to catch the high-speed train to Edinburgh, or else fly. Bath is relatively convenient to Heathrow, so maybe flying is your best option. So, if you flew on Day 6, you could spend Day 5 in Salisbury and use the bus to visit Stonehenge and Old Sarum.
As another plan, can you shift Edinburgh to Days 10-11-12 and then fly from Edinburgh to London? That would give you more days to make your way north and east from Bath by train. You could go by way of Cheltenham and get a taste of the Cotswolds, then transfer northeastward to York, followed by Durham. A day's slight detour would take you to Haltwhistle to see Hadrian's Wall, then on to Edinburgh.