"Do you mind if I make my own?"

You see, Brits have never understood the American cocktail - a kind of generic slam at our national pre-dinner indulgence.

"What are these strange concoctions?" they'll ask. To make matters worse, some Brits, like their continental rivals, think a martini means a few ounces of dry or sweet Italian Martini & Rossi vermouth.

To obfuscate the whole martini ritual in London, add the problem of vodka vs. gin. A gin base is essential for a pure and traditional martini. But what American martini connoisseurs now like is a variant white liquor that is somewhat tasteless and bears no resemblance to gin beyond color - namely, vodka. Tell that to a British bartender, and it's like throwing a spit ball to a batter.

But it's not quite that bad these days - with almost four million Americans visiting London each year and, in the process, providing London bartenders with a bit of martini education, if not a frown if the mixing is botched. Not a little help has come from James Bond who has globalized the martini in film after film with an infidel's touch

("Shaken, not stirred").

But the real impetus for a fundamental improvement in the London model of the American martini has come from sheer economics: most luxury hotels - and even less pricey ones - depend on American corporate travelers for roughly 15% to 40% of their business. And the bulk of those Americans come from the East Coast, particularly New York, and the West Coast, mostly L.A. - territories where drinking a martini has more cachet than downing a flute of Dom Perignon.

Besides, with London's drinking laws so unruly, the only sure place to sip a martini is in a hotel. For a general manager of a high-profile London hotel to receive a complaint about a martini is tantamount to giving the bartender a pink slip. In a few cases, the hotels now have American experts or "bartenders with international experience" minding the store to ensure compliance: a whiff of dry vermouth, a superior vodka like Absolut, a twist of lemon (in the glass or on the side) or olive, and a martini glass pleasantly iced.

The only drawback is the nearly prohibitive price of a London martini (or almost any other cocktail) at today's exchange rate. Double the American price, and you'll be close.

One of the earliest masters of this art form was the bartender at the American Bar off the lobby in the Savoy Hotel (on the Strand). And that wateringhole is still a great repository of martini-making lore. The going rate is $15.75.

Another hotel heavily inhabited by American travelers who know a martini when they sip one is the Stafford Hotel (in St. James's Place), a cozy, picturesque retreat with an American Bar wallpapered with sports and collegiate caps, bunting and photos. In the same league is the St. George's Bar at Brown's Hotel (Albemarle Street off Piccadilly), which produces a thoroughly Americanized martini despite its reputation as the perfect setting for 4 p. m. tea and scones. Both hotels price their martinis at about $12.

For more action while indulging, head for the hotels along Park Lane. Drop in at the swingin' Bar at the Dorchester Hotel, where there's live music to accompany a very mean martini. Down the avenue is the Hilton on Park Lane's penthouse Windows on the World - with a panoramic view of Hyde Park and memorable martinis. At the Four Seasons (near Hyde Park Corner) you can get a good martini anywhere on the premises - even through room service. Park Lane cocktail prices? Mostly in the rarefied $15 category.

Here are some other noteworthy American-oriented hotels where the martini, in its world-class form, thrives: the Hyatt Carlton Towers, London Marriott Grosvenor Square, May Fair Inter-Continental and Sheraton's Park Lane and Park Tower.

Designer and restaurateur Terence Conran's string of hot bistros, from Quaglino's (16 Bury Street near Piccadilly Circus) to Le Pont de la Tour (on Butler's Wharf near Tower Bridge), are adept at concocting American martinis - if only to keep customers tranquilized while waiting for a table.

La Gavroche (43 Upper Brook Street, near the U. S. Embassy), a temple of haute cuisine, would hardly survive without producing a superb dry martini.