Britain's unions staged a major protest against Prime Minister Margaret Thatcher's government today, with work stoppages and rallies around the country in support of pay demands by employes of the National Health Service.

The Trades Union Congress, organizer of the action, said millions of its members on the docks and in mines, in schools, newspapers, some government offices and scattered factories, staged walkouts ranging from one hour to a full day. A crowd estimated by the TUC at 80,000 marched through London to Hyde Park to hear union and Labor Party spokesmen demand higher pay for the health workers.

The nationwide protest, probably the most broad-based union action in recent years, was designed as a show of strength and solidarity by the 11 million-member TUC. Its purpose was to back nurses, ambulance drivers and other health workers in their bid for a 12 percent wage hike. The underlying objective was to challenge the economic policies of the Thatcher government, which the unions blame for Britain's 14 percent unemployment.

Today's activity demonstrated that union leaders can turn out a significant segment of the country's work force despite the success of Thatcher's hard line with other recent strikers, such as railway engineers. What is not clear, however, is whether this "Day of Action" marks a resurgence of the sort of labor unrest that brought down British governments in 1974 and 1979.

TUC President Frank Chapple, a union moderate, projected a limited goal for the stoppages. He said they "were a show of spirit" in specific support of the health workers' grievances. The government's latest offer is for a 6.5 percent increase for nurses and 5 percent for others, retroactive to last spring when the dispute began.

The health workers' claims have generated considerable sympathy because many of the 600,000 employes of the service are considered poorly paid.

Militant unionists, however, see today's actions as a wedge toward a broader challenge of Thatcher policies. On a trip to the Far East, Thatcher said no amount of protest would lead to a higher pay offer.