Does your week actually mean 7 days on the ground excluding arrival and departure days? It sounds like you may only have 5 days net.
Winter - stick to the cities such as York, Chester, Durham or Edinburgh. It’s a pity to waste half a day back tracking from Edinburgh to London. Take the train.
Summer - hire a car and get to see the countryside, coastline and smaller towns. I would try to focus on one area, rather than trying to cover half the country and spend hours driving. So many options! Take temporary membership of the National Trust for your dose of history.
Kent/Sussex - quick to get to from London. Visit Dover Castle, Canterbury, spend a night in Hever Castle, many world class gardens such as Sissinghurst and Great Dixter, Rye for cobbled street quaintness, Lewes for another castle and Anne of Cleeves House and the gorgeous local villages of Alfriston, Jevington and Rodmell.
Norfolk/Suffolk - gorgeous villages and coast around Cley next the Sea, take a boat trip out to see the seals, pretty National Trust properties such as Lavenham, Long Melford, Felbrigg and Blickling. Stay in The Tower at Blickling Hall. Look out for the Queen in Sandringham. Take a boat trip on the Norfolk Broads.