Britain's system of free, comprehensive medical care, pride and symbol of this country's commitment to the welfare of its people, is going through its most troubled period since it was founded with high hopes 34 years ago.

Soaring costs, an aging population, months of labor trouble and a belief that Prime Minister Margaret Thatcher's government favors an increase in private medicine, have led to what is widely seen as a severe crisis of confidence in the health service's ability to meet the public's needs.

"It's sad," said Gordon Harrhy , a regional administrator here. "We have raised peoples' expectations . . . and then we're not able to achieve the results they have been looking forward to."

Waiting lists of two years or more are increasingly common for orthopedic, ear-nose-throat and gynecological surgery. There are reports throughout the country of cutbacks in hospital space, the closing of outpatient clinics and restrictions on advanced treatments such as kidney dialysis, marrow transplants or the insertion of pacemakers. Purchases of new equipment are being postponed or canceled, and hospital maintenance is being scaled back.

"In bluntest terms," observed J. Keith Moger, director of the vast 11-year-old University Hospital, Wales' best, "the cuts mean that many patients who go on waiting lists will never come off."

In a sense, health care problems in the countries of the industrialized West are a product of success. Demands on facilities have risen sharply, and medical science has developed spectacular new means of helping the sick, but at an expense far outstripping society's ability to pay. As one of the leading welfare states, with perhaps the foremost nationalized health system and the deepest economic recession, Britain's plight is particularly difficult.

The extent of the pressures comes through vividly in Wales because it has a smaller health budget and an administrative structure that is separate from England's but with all the same problems. A recent confidential estimate by senior local officials, which caused a furor when it was leaked, predicted "an unprecedented squeeze in 1983-84." The report called for the "most vigorous and determined attack on costs and control of manpower." To outraged union leaders, that meant fewer jobs and reduced medical care.

"We've already got a sick economy," said Stewart Barber, a Cardiff organizer for the National Union of Public Employes. "If we dismantle the National Health Service, then we'd have a sick population."

In fact, despite restraining expenditures as much as possible, the Thatcher government is spending more in real terms on health than any previous British government. The health service's share of overall social spending has inched upward annually. Figures announced recently for the next fiscal year maintain the pattern of increases at about 12 percent over inflation.

But just to stay even with mounting requirements, in the opinion of many health experts in and out of government, a boost of around 2 1/2 percent is necessary, but this conservative administration plainly will not provide it. The main reason costs are running so much higher are these, experts say:

* There are now 3 million people over the age of 75 in Britain, a figure that increases by 75,000 a year. Their health care, it is reckoned, costs six or seven times more than that of working-age people. Just to provide for the aged, 30,000 extra hospital places will be necessary by the end of the century to maintain present services. Higher standards also are being sought in the care of the mentally ill and handicapped.

* Advances in science and technology are offering the prospect of expensive treatments not available previously, and under the British system, they will be given for free. Among the longest waiting lists, for instance, are those for hip and knee replacements, operations that only began on a large scale a few years ago. "The more we can do," said one health official, "the more we are asked to do."

* Health workers are demanding wage increases that are greater than the amount the government has forecast in its budgets.

For seven months, the system has been hit by regular strikes and slowdowns by nonmedical staff, ambulance drivers and some nurses. This has lengthened the delays in treatment and means a pay settlement that will have to be partly met by money earmarked for other purposes. With almost a million employes, the health service is the biggest employer in Western Europe and about twice the size it was in 1948.

Also blamed by some are the high drug prices charged by multinational pharmaceutical companies (most patients now pay a fee for medicine of about $2 per prescription) and the inefficiencies of what critics say is a bloated health service bureaucracy. Moreover, many of Britain's facilities are old, and replacing them means massive capital expenditures. As one official put it, "The fabric of the hospitals, their physical presence, is deteriorating."

How all of this will translate in practical terms for Wales, based on expected spending well below the optimum, is "devastating," said Albert Huish, a longtime member of the community health authority for Cardiff. With what amounts to minimal-growth budgets, Wales will continue construction of five new hospitals, mostly to replace existing ones.

There is no way these "claims for central funding" can be met without "biting into . . . baseline revenues," the government's confidential estimate said. At University Hospital, for instance, a general surgical ward with four nurses will now have to function with three. Plans to upgrade a cardiac ward have been shelved. A day-old baby had to be rushed to London by ambulance because lifesaving equipment was not available in Cardiff.

Last year the hospital cut 80 employes out of 2,000, mostly by attrition. More will certainly go this year. "We are going to have to examine very much more carefully what everybody does and how we spend our pounds and pence," said Moger, the hospital's director. "That in itself is not catastrophic. But having said that, I think there is going to be a considerable difference as to service -- both in standard and quantity -- we can provide."

Similarly serious expressions of concern are heard at every level of the health service, from ward nurses and neighborhood general practitioners, from local hospital administrators to senior officials of the Health Ministry in London. Summarizing the sense of alarm, the respected Times Health Services Supplement last month published an open letter to Thatcher that declared: "We are not suggesting that you have deliberately set out to destroy the National Health Service, nor that you want it to decline, but the reality is that current economic circumstances are threatening the service with destruction unless your government does something to rescue it."