Where there are alternative spellings I have given the most commonly used.
Britain, UK etc: Great Britain is the island of England, Wales and Scotland (excluding the Isle of Man and the Channel Islands); the United Kingdom is England, Wales, Scotland and Northern Ireland (excluding the Isle of Man and the Channel Islands); the British Isles is a geographical term for the whole group of islands including Ireland, the Isle of Man, the Scottish islands and the Isles of Scilly. For practical purposes Britain is usually good enough. I would prefer Britain in the first reference in copy but UK is fine as an alternative and is most useful for headings. The Channel Islands are not part of the British Isles as they are closer to France. The Channel Islands and the Isle of Man are dependencies of the British Crown (or Crown dependencies) and organise their own internal affairs, with the British government responsible for defence. The Channel Islands are divided into two self-governing bailiwicks, Jersey and Guernsey.
London: In line with my recommendation to use as few caps as possible (see If the Cap Fits section) I would use lower case with hyphens where necessary, thus: south-east London, north London. Where possible give a locality too, eg Chiswick, west London. Well-known London areas such as Kensington, Mayfair, Hampstead, do not need to be followed by London. If the location name involves a compass point, such as East End and West End, give it caps.
the outer London boroughs present a particular problem. They mostly cover a combination of London postal districts and county addresses. For example, the London borough of Bromley includes SE20, which covers the built-up areas of Penge and Anerley, and Orpington, which is on the edge of the countryside. Placing Orpington in south-east London seems all wrong to anyone who knows the area. I think the best solution is to call anywhere within a London postal district ‘London’, and for everywhere else use the county featured in its postal address, eg ‘Orpington, Kent’. The only snag here is that Middlesex, a former county and postal address, no longer exists except in the names of various organisations. All you can do is place the various locations which used to be in Middlesex in north or west London.
rest of UK: Cap up obvious geographical entities, such as the South, the North, the South East (no hyphen) and accepted usages such as the West Country, the Yorkshire Dales, the Lake District. (The reason for the difference from south-east London etc is that the South East is a proper name while south-east is used as an adjective.) Even when the name is a plural, such as the Yorkshire Dales or the South Downs, it should be treated as singular eg ‘The Brecon Beacons is an area of outstanding beauty’.
counties: I think these look better spelled out in full, such as Buckinghamshire not Bucks. If a town is well known you do not need to give the county, and the usual rule of thumb to decide if a town is well known is whether it has a football league team. Use your judgment.
name changes: We would not dream of using Siam for Thailand or Ceylon for Sri Lanka, but we hesitate over more recent name changes such as Mumbai for Bombay, Myanmar for Burma and Chennai for Madras. It's just a matter of how long the new names take to become familiar. I suggest that for the first year or so after a new name is brought in, you use the new name with the phrase ‘formerly known as xxxx’ in brackets. If after that time the new usage seems to have become accepted, don’t bother with the old name. If on the other hand it does not seem to have caught on, continue with the qualification.
compass points: Should usually be lower case and with hyphens if necessary, eg ‘He is believed to be heading north’, ‘the ship sank six miles south of the Equator’, ‘a village south-east of Woking’. There are bound to be exceptions, such as ‘Go West, young man!’
home counties: a slightly nebulous term for the counties surrounding London. Those which touch London are obviously on the list, viz Kent, Surrey, Hertfordshire and Essex, but the neighbours Berkshire, Buckinghamshire and Sussex are usually included, and occasionally Bedfordshire, Oxfordshire and Hampshire. It is less of a geographical than a social term, used to describe the traditional domain of the middle and upper classes.
shire counties, or shires: counties which existed before metropolitan authorities covering large towns were introduced in 1974. Most, but not all, end with ‘-shire’ and all have a county town after which most, but not all, are named. As with home counties, the expression is not so much geographical as social, describing a real or imaginary ‘Middle England’.
America: Strictly speaking this covers the continents of North and South America. If you are talking only about the United States, and it is not obvious from the context, this should be mentioned somewhere. US/U.S./USA: The Daily Mail insists on full points in U.S., but this dates back to a time when it frequently used caps headings, and US could be misread for the word ‘us’. I recommend ‘US’ without full points in copy and headlines, except in caps headlines, and USA without full points.
Beverly Hills, not Beverley
Brasilia is the capital of Brazil, not Brazilia.
Bridgwater, Somerset, not Bridgewater
Broadmoor is a hospital, not a prison, and inmates are patients, not prisoners. The same applies to Ashworth Hospital, Sefton, Merseyside, and Rampton Hospital, Nottinghamshire.
Caribbean, not Carribean
Channel Islands: Guernsey and Jersey do not need to be accompanied by ‘in the Channel Islands’. The smaller islands of Sark and Alderney probably do, but a better construction would be ‘the Channel Island of Sark’.
Costa Rica: a Central American country; Puerto Rico: a Caribbean island
Disneyland, California; Disney World, Florida (officially named Walt Disney World); Disneyland Paris (both words in title, no comma).
Dominican Republic is the neighbour of Haiti on the island of Hispaniola, not an island on its own, while Dominica is an island, the dividing line between the Windward Islands and the Leeward Islands in the Caribbean.
Dunbartonshire (with an 'n') is the name of two Scottish council areas, East and West; Dumbarton (with an 'm') is the name of the town in West Dunbartonshire.
East London is a town in South Africa. If you want to talk about part of London, it is east London or the East End.
Hinckley is in Leicestershire, Hinkley Point is in Somerset.
hyphens or not: Newcastle upon Tyne; Stoke-on-Trent; Henley-on-Thames. If you are not sure look it up on a council website.
Lake District: Windermere and all the other ‘meres’ do not take the prefix ‘Lake’, because ‘mere’ means ‘lake’. The same applies to Derwentwater and all the other ‘waters’. The only lake in the Lake District is Bassenthwaite Lake.
Londonderry in Northern Ireland is favoured by unionists, Derry by nationalists. Officially the city and county are called Londonderry, and the local government district is called Derry.
Manchester/Salford: Salford is Salford, not Salford, Manchester, or Salford, Greater Manchester. It is a city in its own right, adjacent to Manchester, with a cathedral and a large university. Hence the advertising slogan for Boddington’s beer: ‘Ale of two cities’. Annoyingly, the Lowry Hotel, named after Salford’s son L S Lowry, brands itself as being in Manchester.
Middlesbrough, not Middlesborough
Philippines: A male from the Philippines is a Filipino and a female is a Filipina. Both are nouns and adjectives, so you could say 'He is a Filipino singer', 'She is a Filipina', or vice versa. The adjective and noun that covers both is Filipino.
Sahara: Purists say there is no need to add ‘desert’ because Sahara means desert in Arabic. For those not fluent in Arabic I don’t see the problem with adding ‘desert’. On the other hand, could anyone reading the word Sahara wonder if it means a mountain or a sea?
Shepherd Market, Mayfair, not Shepherds or Shepherd’s
Siena in Italy; but Sienna Miller, actress, and burnt sienna, colour
St Albans, Hertfordshire: no apostrophe
St Andrews, the Scottish university and golf course: no apostrophe
Stansted airport, not Stanstead
Tehran, not Teheran
Westmorland was a historic county now covered by Cumbria, hence the Westmorland and Lonsdale parliamentary constituency and the Westmorland Gazette; US General William Westmoreland was prominent in the Vietnam War.