score:7
You will have to modify your itinerary. Visiting all the places in your first group in a week might be possible if you went by air and taxi, but you would have perhaps an hour in each town; not really worthwhile.
Stratford, Bath and Stonehenge (and Oxford) are close enough to combine into one trip of three or four days. York and Nottingham (and perhaps Liverpool, travelling by train) could be day-trips from Leeds, as Cambridge and Brighton could from London. Travelling from Leeds to London, via Nottingham, would be a long day. Edinburgh and the Lake District (as also John O'Groats, Lands End and Anglesey) are out of reach on a trip such as you propose.