Don't forget Salisbury Cathedral on your trip to the South-West. Try to attend choral evensong at one of the great cathedrals with a serious choral tradition (it doesn't matter if you are Christian or not, it's still a great experience). All such cathedrals put their service lists up on the website so you can see on which days they are singing. St Paul's and Westminster Abbey in London, Canterbury, Winchester, Salisbury and Wells, and don't forget Christ Church and New College, Oxford. The musical standards at Westminster Cathedral (Roman Catholic) are very high and a sung Mass there is a memorable experience.
When you are in London make sure to visit Harrod's Food Hall, and look in at the Victoria and Albert Museum and Brompton Oratory at the same time. If the weather is nice the walk along the Thames from Tower Bridge to Westminster Bridge is fascinating. Go to some 'period' Shakespeare at the Globe Theatre. Maybe take in the National Theatre or National Film Theatre. The South Bank concert halls are ugly, but there are great concerts in the Royal Festival Hall and Queen Elizabeth Hall. The Barbican is a funny place but it, too, hosts high quality concerts.
If you are doing the tourist sights of Buckingham Palace (probably only the art gallery open), Changing of the Guard,St James's Palace, Horse Guard's Parade make sure to look into Berry Brothers and Rudd historic wine merchants at the bottom of St James's. Get them to bring you up a bottle of something special from the cellar and have yourself weighed on their coffee machine. Then you can continue up Bond Street and take in some of the great shops such as Aspreys - and don't forget the Burlington Arcade, or Fortnum and Mason in Piccadilly. The National Gallery is justly famous, and the Tate Gallery specialises in British art - and has offshoots in London, Liverpool and St Ives. The river cruise from Westminster Bridge up to Greenwich and back is well worth doing, leaving yourself time for an explore of Greenwich, the Palace, the park, Royal Observatory etc. Kew Gardens are absolutely special, even on a rainy day. The great palace at Hampton Court is easily accessible by train and quite splendid. Leave time for the gardens and maze. The Cabinet War Rooms are very interesting if that sort of thing appeals (ditto Imperial War Museum) but I think for anyone interested in the great wars the RAF Museum at Hendon is a must - it's not only British planes, there's a very moving exhibit of a Flying Fortress beside a control tower, as if about to set off to bomb Germany. Will it come back? There are German planes, too.
I have a particular fondness for the downs running through Oxfordshire, Wiltshire etc with their prehistoric burial grounds, ancient pathways, Stonehenge, Avebury and so on. Hungerford, Marlborough, Newbury are attractive towns with decent shops and the little roads around there are fascinating. Should you choose to drive to the West Country I recommend a detour to the Wiltshire village of Mere. The old brewery there houses one of the best rural family-run wine shops in the country, Yapp Brothers. They are specialists in the wines of the Rhone and Loire (and are having a sale of some of their archive wines at the moment - look on their website), but they do stock wines from other regions with excellent pedigrees such as Schleret of Alsace. They are by no means the cheapest, but you could pick up a bin end of something rather nice to drink in your hotel room in the evening.
If you are going to be out of London at a weekend you might Google farmers' and artisan markets in the area. These are not only foodie haunts, but there are usually craft, antique an bric-a-brac interests as well. You'll have the opportunity to taste all sorts of hand made cheeses, pates, local beers, local honey and you could partake of a few Gloucester Old Spot sausages or a water buffalo burger for lunch as you stroll round.
As well as the big country houses, the West Country is littered with wonderful churches, including the famous one once buried in sand in the middle of St Enodoc golf course, in the graveyard of which is buried the poet Betjeman. I think the regional tourist boards do a pretty good job of advertising all that may be of interest on their websites and they are very good about links to other relevant web sites, whether you are interested in Roman remains or real ale pubs.
Feel free to IM me if you would like further information if you have specialist interests. I may be able to help. If I don't I usually know somebody who does know.