At St. Paul's Cathedral, more than 30,000 people traipsed through the aisles and clambered up to the Whispering Gallery each day last summer.

At Buckingham Palace, crowds of 20,000 of flocked daily to see the Changing of the Guard while up to 200 tour buses - stretching almost a mile - arrived to find parking space for only 92.

On three days - July 18, 19 and 20 - there wasn't a hotel or guest house vacancy anywhere in the city. That had never happened before.

It isn't easy to overwhelm a city of 7 1/2 million people, but this year the world's tourists finally managed it with London. In the early 1960s, London received about 2 1/2 million foreign visitors each year. By 1970, the number had doubled. Now, approaching 9 million, it has almost doubled again. Add in 11 million domestic visitors and it grows to 20 million - almost three times the city's population.

In short, London, in spite of its grim food and hopeless weather, has become the most poular tourist spot in the history of the world.

This, of course, means big business. Foreign visitors spent close to $3 billion here last year, making London tourism the third biggest industry in Britain - bigger even than North Sea oil. But this popularity is also exacting an enormous toll on the city's monuments and facilities.

At Westminster Abbey, where it has been calculated that the maximum number of visitors that can be handled at any one time is 700, the crowds at peak hours (at about 11 a.m. on any July or August weekday) have exceeded 3,000. A rigid one-way system already has been imposed, but even so the abbey's dean has said that within two years - unless the crush eases - it is quite possible he will have to turn sightseers away from his doors.

Nowhere is the problem felt more acutely by tourists than at the hotels, where summer vacancies were scarcer than sunshine.

Aided by government grants, businessmen erected 125 hotels in London between 1969 and the 1974 recession, but it was not enough. In fact, it has been estimated that if all those who wish to visit the city in 1980 are to be accommodated, London needs 220,000 more hotel beds. With building costs of up to $75,000 per room, only a few dozen are being added each year. As a result, many visitors this summer had to travel to Reading or Oxford to find a place to sleep.

London hotels are no longer the bargain they once were. Inflation has brought prices up faster than the falling pound could keep them down, so that most now charge about double the rate of two or three years ago.

True, the average cost for a double room in a modest hotel (for instance, at the Royal Bayswater, 122 Bayswater Rd.) is only about $15, and it is still possible to find accommodation for one-half to two-thirds that in the guesthouses that proliferate in Kensington and Bloomsbury. Prices invariably include a substantial (if sometimes indifferent) English breakfast.

But at the larger hotels, Prices have skyrocketed. At the Hilton, 22 Park Lane, rates are from $55 to $62 for a single and from $70 to $85 for a double, depending on the season.

A few other centrally located hotels and their prices: Barkston Hotel, Barkston Gardens, $18 single/$24.50 double; Browns Hotel, Albermarle Street $34-$39/$50-$58; Cavendish Hotel, Jermyn Street, $36.50/$50; Grosvenor House, Park Lane, $39/$59. Prices are for low season; after April 1, you can expect to pay $5 to $10 more.

The London Tourist Board crisply denies that the situation has gotten out of hand.

"London has more than enough attractions to accommodate the growing numbers," insists Robert Chenery, the board's manager of resource development. "The only problem is to get the attractions shared more equably among the tourists."

Others are less certain. Sir George Young, a former tourism official for the Greater London Council, the city's governing body, believes London reached its saturation point some time ago. He points out that on a basis of tourists per 100 residents, the city is far more crowded than many of the world's traditional tourist haunts.

"The annual number of visits to the U.K. is probably not in itself too high," he has written, "but too many visitors are in the same place at the same time."

Indeed, of all visitors to London, 93 per cent go to Trafalgar Square, 85 per cent to Westminster Abbey, 83 per cent to Buckingham Palace and 82 per cent to the Tower of London. Three-quarters of them never even leave the city.