After the wettest June and coldest first half of July on record, summer has suddenly arrived this week in Britain and the rest of northern Europe. Temperatures have been rising to the sizzling high seventies.

For some Londoners, it takes just a few days of this sort of heat wave to make pleasant memories of the seemingly endless weeks of gray skies, intermittent rains and temperatures that stayed down around 60. But the British tourist industry hopes that the belated emergence of the sun will bring some recovery of its badly depressed trade.

Tourism and spending by those tourist who are here have slumped for the second straight year, aggravating Britain's other economic problems. Hotels in prime tourist areas of Scotland, Wales, southwest England and the south coast beach resorts are only half full near the peak of their season. A big drop in summer trade is threatening many London shops and theaters that had depended on the booming tourist trade of the 1960s and 1970s to tide them over until the Christmas holiday period.

Tourist officials blame a variety of factors, including the bad weather of the early summer, the recession throughout Europe and the United States, and the rapidly rising relative cost of vacationing here.Hotel and restaurant prices have been driven up by the 20 percent inflation rate. And the high exchange value of the pound, inflated by Britain's North Sea oil income and high interest rates, have made the cost of everything here about double prices in the United States.

As a result, the number of American tourists in Britain has fallen by more than 10 percent each of the last two years (compared with annual 10 percent increases most years before that), and Britons and other Europeans are passing up vacations here to take advantage of the cheap air fares across the Atlantic and the bargain prices of hotels, meals and souvenirs in once-expensive America.

Big spenders, especially from America and the Middle East, are scarce here this summer. Harrods, the well-known department store here, had resorted to bargain sales to keep its revenues rising after the proportion of tourists among its customers dropped from 40 to 20 percent. More German and French than English can be heard among tourists at the Tower of London, and the once-ubiquitous Arab oil sheiks and their families are missing from Harrods and nearby Hyde Park.

Belatedly, British tourist officials are intending to do something about this after years of sitting back and letting the tourists and money roll in. An "I Love London" promotion campaign is being mounted with buttons for the tourists who made it here this summer and an advertising blitz in Europe and America this winter.

Prices are being discounted at some medium- and high-priced hotels, which cost from $70 to $300 and more a night. One hotel chain is offering cut-rate prices to Americans flying on British Airways if they agree to make their hotel reservations just before getting on the plane to Britain. Others, like Hilton, are cutting prices for package tours for the first time. o

There is also a campaign to clean up the streets of the entertainment and shopping districts of London's West End, which has become noticeably down at the heels. The theaters of the West End, London's Broadway, also plan to sell unbought tickets at half price from a kiosk on Leicester Square, just as New York theaters do in Times Square.

London's celebrated theaters have suffered noticeably from the drop in tourists and the effect of inflation on leisure spending here. A recent Times newspaper survey of London's 40 West End theaters found seven shut and the rest only two-thirds full on the average during what would have been a busy summer week a few years ago.About 87,000 tickets worth nearly $1 million in revenue went unsold that week.

Producers and theater owners are complaining that the London stage may not be able to survive in its present form with revenues falling and costs escalating. Already, a disproportionate number of West End dramas and musicals have been brought here as sure-fire box offices successes from New York or the subsidized regional theaters of Britain to minimize financial risks. And one of the best attended London theaters, the three-stage National Theater in its new home on the South Bank of the Thames River, is heavily subsidized by the government.

Plays and musicals that are breaking even or making money for their backers depend primarily on nostalgia (revivals of the "King and I" with Yul Brynner and "My Fair Lady"), outstanding performances by big stars (Glenda Jackson in "Rose," Tom Courtenay in "The Dresser," and Paul Scofield in "Amadeus") or on rave reviews of once-in-a-generation productions (a bigger-than-life, two-part, 8 1/2-hour dramatization of Dickens' "Nicholas Nickleby" by the Royal Shakespeare Company and an electric new version of "Hamlet" with the Royale Shakespeare's brilliant young Christoper Pryce).

Many others are in trouble, "Jesus Christ Superstar" is finally scheduled to close, and "No Sex Please, We're British," London's longest running comedy, is failing to fill half its house. b"Chicago," a hit musical in New York acclaimed by critics here, was kept going only after its cast volunteered to take pay cuts and the actors' union invested money in the production.

Theater owners, who avoid discussing details of their finances, and newspaper critics, who have tried without success to probe into them, are divided about how serious the trouble is. With theater-going by tourists clearly well down this summer, the theaters will be campaigning hard for local trade this autumn when a number of new plays -- including more revivals and transfers from Broadway -- are scheduled to open.

Meanwhile, with fewer tourists on the streets and slightly less traffic, the summer season goes on here with the royal race meeting at Ascot country house parties and the queen's garden parties in the vast garden of Buckingham Palace.

"It's just like the Titanic sinking while the band plays on," said one of the many London taxi drivers who are complaining louder than usual this summer about the decline of Britain. "The country's sinking," he said as he skirted the traffic jam created by 6,000 people on their way to one of the queen's garden parties, "but the party goes on."