Posh Britain; DOING VERY WELL INDEED

THERE'S one corner of Britain that's apparently doing very well indeed. It's high upstairs Britain, posh Britain, quality Britain. One or two establishments I enjoyed particularly are:

Claridge's, Brook Street

At its peak is Claridge's, whose suites act as stately homes away from stately homes for the very royal and the very rich. The footmen are straight out of Cinderella and there seems to be a chef for every imaginable course of a royal banquet. Suites have rooms (including superb dining rooms) that would not disgrace a palace, bathrooms that could house a family of four, and there's not a chipped marble tile or a tiny speck or a pulled thread anywhere. Part of the hotel is authentic Georgian, part authentic art nouveau, and it is easy to see why it is under a preservation order. I would call it a museum piece if that doesn't sound too stuffy.

In the old days when so many of Claridge's guests arrived by ocean liner, baggage trains would reach right around the hotel. Correct dress was essential - after all ''you would dress up if you were dining with the Queen, wouldn't you?'' asked Mark Richards, assistant resident manager. Claridge's can provide anything its guests ask for, he said.

''A camel?''

''We could tell you where to get one . . . but please don't bring it into Claridge's.''

For an opportunity to dream about high life for half an hour or so, try ordering an orange juice in Claridge's some time between 7:30 and 10:30 and listen to sweet music. Hatchard's Bookshop, Piccadilly

One of the pleasant things about Hatchard's is that a book bought there costs no more than one bought anywhere else. Being treated like a VIP and partaking of a little of London's history is all free.

When the shop began in the l8th century, it wasn't only a bookshop, it was a kind of club concerned with reform (William Wilberforce who worked to abolish the slave trade was a member). On June 30, 1797, the founder wrote in his diary , ''When I commenced business I had less than five pounds, but God blessed my industry and good men encouraged it.''

On a less high-minded level, the sermons Hatchard's sold were written by hand so that no member of his congregation would suspect his minister of using ready-made sermons.

Nowadays service is Hatchard's strong point. For instance Peter Giddy, director of the company today, says that if a customer reports a package of books has gone astray, it is immediately replaced. ''We replace first and then investigate at the post office.''

Mr. Giddy's unfavorite customers are the ones who tell their children, '' 'Don't touch the books.' Now why do they say that? They are my books. Books are meant to be touched.'' Fortnum & Mason, Piccadilly

Back in 1707 a shopkeeper, William Fortnum, and a royal footman, Hugh Mason, got together and opened an establishment in Piccadilly that has never quite lost its regal aura. It sells items like ''Partridge with foie gras,'' delivers a quarter of a pound of coffee every week to a customer in London, and once flew a packet of potato chips to a hungry customer in Los Angeles. The shop also sells clothes, furniture, and antiques, but it is food that maintains the company's fame.

''Elevenses'' (mid-morning snack) at the Garden Restaurant is a good place for shopper-watching, VIP-spotting, and talking (a lady at the next table told me that when her mother was a servant at one of the big households nearby she used to pop across to do the shopping in her print dress). Liberty's, Regent Street

Liberty's opened in 1875 -- a good example of supply not only meeting but increasing demand. While the art world (the pre-Raphaelite brotherhood) with its emphasis on soft, easily draped, beautifully colored fabrics was making stiff, harshly dyed clothes unpopular, Arthur Liberty was importing fine materials -- first from Japan and then from other Eastern countries. He used his own specially developed dyeing process to produce the delicate pastel shades that have made the firm famous ever since. Fabrics are still Liberty's specialty, but its carpets, ceramics, and jewelry make it marvelous browsing ground for the shopper. And, just to show what quality and ingenuity can do, Liberty's now export fabrics to Japan. Maids of Honour Shop, Kew

One day Henry VIII noticed that his maids of honour were eating something quite delicious, so delicious that he ordered the cakes should never be eaten outside the royal circle. He put the cook in a cottage on the royal grounds and never allowed her out. But at the end of his reign a Mr. Beillet got hold of the recipe and began selling the cakes, named ''maids of honour'' in honor of the maids of honour. Now the great-great-grandson of Mr. Beillet's head chef, Peter Newems, is making them (it takes 24 hours) and selling them in a small pastry shop in Kew where the food looks - and tastes - as if it had been prepared for an ''Upstairs Downstairs'' banquet. No wonder. One of his pastry cooks has been with him for 63 years. The shop, located at 288 Kew Road, Kew Gardens, Surrey, is also open for lunch. Choice is limited to make sure the food is always fresh. James Lock & Co., St. James's Street.

Ever since l676 Lock's has been supplying hats to kings (two Edwards among them) and gentlemen (including Field Marshal Montgomery, Charles Chaplin, and Nelson, who favored cocked hats with green silk eye shades). It was Lock who invented not only the bowler (or coke) but a terrifying contraption called the ''conformateur'' for plotting the shape of a customer's head. The first bowler was made for William Coke of Norfolk (nephew of the Earl of Leicester) who wanted a hard domed well-fitting hat to protect his gamekeeper's heads from falling (or even thrown) objects. When the first hat was ready, Mr. Coke tried it on (it fit) trod on it twice (it survived the test), paid 12 shillings, and started a fashion. If you order a bowler today, Lock's will measure your head with their conformateur and 16 weeks later a perfectly fitting hat bowler will be ready. It will have spent its first eight weeks in the Midlands and all that time it will have been wet. (Midlands water is softer than London and preserves the color).

Of course, bowlers are certainly not the only hat Lock's makes - in fact it's hard to imagine what kind of headgear you couldn't buy there.