Hmmm, nature and history in October...The weather can be quite bad (as in wet and miserable): you need some flexibility so that you can switch from indoors to outdoors depending on the weather, and good rain gear. Fortunately, there is so much to see in this country in the way of castles, mansions and stately homes, cathedrals, to keep you busy even if it rains for the whole two weeks of your stay.
Although the country is quite compact, there is so much to see that it makes sense to concentrate on a couple of areas rather than rush around doing a Top10 bucket list.
The Lake District is beautiful for serious walking. West Sussex is more gentle but also good for walking (lots of trails around the South Downs, the famous white cliffs between Seaford and Eastbourne, cute villages everywhere like Amberley, Petworth), also the New Forest near Southampton (incl. Beaulieu or Buckler's Hard). Also the scenery in Wiltshire (around Marlborough, Devizes, etc.), the coast in Dorset and Devon, Cornwall too (although it is further, it takes the best part of a day to get there from London)... So pick one or two of these areas.
You will need several days in London, indeed you could spend a week there and only scratch the surface. Despite what some of the haters here suggest, it is an attractive city (much improved from the 60-70s) with a lot to offer the tourist. You must see the classic tourist area, essentially between Knightsbridge and Covent Garden: work your way from the Victoria & Albert museum, pop into Harrods, see Buckingham Palace, the Mall and St James' Palace/Square/Park, Westminster Abbey, Parliament, Whitehall, the London Eye, Trafalgar Square and National Gallery, Soho and Covent Garden. Then you must go to the area around the Tower of London, and see St Paul's and the Tate Modern on the way. Maybe take in a concert at the Southbank Centre: good views of the Thames from the bar terrace upstairs. If you have time, push on to Greenwich for the Cutty Sark clipper museum and some fine Ch. Wren architecture. The river bus can be a good way to get around.
Just outside London: if you're interested in history, you must see Hampton Court and/or Windsor. You might consider getting a yearly pass for Historic Royal Palaces, which will get you into Hampton Court, the Tower of London, Kensington Palace, Kew and a few others: it can work out cheaper even if you don't return. Doesn't cover Windsor unfortunately.
https://www.hrp.org.uk/#gs.tzghc9E If in Windsor, you can also take a look at Eton and see the Thames around Henley and Marlow.
Outside London: Oxford or Cambridge are classics, also Bath. Bath can be combined with Bristol and a visit to Stonehenge (nicely renovated now), Salisbury (nice cathedral) and some of the market towns on the way. But you will need to rent a car (and learn to drive on the "wrong" side, changing gears with your left hand; better still: get an automatic). If you go to Cambridge, Ely, Bury St Edmunds and Norwich can be a good circuit if you want to see cathedrals. Last but not least: Canterbury in Kent (beautiful and historically cathedral in a fairly well preserved town. Lots of tourists, but maybe October will be quieter).
