WHEN PRESIDENT Jimmy Carter comes to Washington, England, in early May to the home of the earliest known ancestors of the first United States President, George Wasahington, it is hoped he will be greeted by members of the Washington Jazz Band.

The Washington Jazz Band is made up of children in the northeastern England area who play fifes and drums in the traditional "Yankee Doodle" manner.

As it is, the young musicians are all set to play - as they have in recent years - on Independence Day, July 4, when the Stars and Stripes is formally raised at Washington Old Hall, the ancestral Washington family home. The scheduled appearance of President Carter two months before that time can provide an added incentive for the youngsters.

There are two otther great attractions bringing the President to that part of England after his arrival in London. First, there is Washington New Town, which British government hosts want him to see. This is one of the planned communities built by Britain in the post World War II years to provide both homes and jobs in an attractive setting for British workers.

The second, a site which any American coming to this area 250 miles north of London would want to see, is Washington Old Hall where, in the 12th century, one William Washington, our George Washington's earliest known ancestor, made his home.

Washington New Town sprawls all over the map. It is about midway between the medieval cathedral town of Durham and the industrial city of Newcastle, to which (as every American knows) one need never carry coals. Times have changed, however, and Newcastle, which the British governmentwants Carter to see also, is no longer the prime coal-producing region it once was. Supplanting coal as economic backbone of the area are diversified, lighter industries dispersed across the countryside that have made possible the development of such communities as Washington New Town.

The vast changes of recent years wrought by the establishment of Washington New Town have not, however, infringed upon the sacrosanct area at its center compromising the medieval village of Washington. And it is there, at the village, that you find Washington Old Hall, the family home. The village of Washington remains unchanged despite the mushrooming suburbia-type homes constructed in the surrounding area.

The people of Britain, through the National Trust, have seen to it that Washington Old Hall, the earliest ancestral Washington family home, has been restored from the physical wreck it was in the early 1950s.

A rented car is the best means of getting to the village of Washington. Fortunately, the drive through the maze of twisting suburban streets of Washington New Town is facilitated by signposts placed at intervals to aid American visitors to the scene during the 1976 Bicentennial Year. Nevertheless, following the direction signals to Washington Old Hall can be a bit of an ordeal.

Some residents of the area give the distinct impression of not being in favor of what has happened there since Washington New Town has been built. Along a road you may hear, as I did, some village matrons deploring what has transpired in their part of England.

"It's all new houses and new highways, dearie," said one woman I encountered. "We're even cut off from our friends and neighbors by highways - strangers i our own home."

But the new highways have not yet slashed up the grounds of Washington Old Hall and are not likely to in view of its sheltered position on the edge of the village. Just off the main crossroad of Washington village lies the entrance gateway. Genealogical records show that this family from 1183 to 1399.

It was sold by descendants in 1615 to the Bishop of Durham. Three and a quarter centuries later, the old stone house came on the market for demolition. A local preservation committee saved it with the help of generous gifts from both sides of the Atlantic.

In 1955, the house was officially opened by the U.S. ambassador and two years later was given to the British National Trust. The Trust, which has a businesslike approach of not taking over any property without any endowment, made a special exception here, in view of the property's historical importance to both England and the United States. To make ends meet, ther pragmatic Trust rents part of the house to the Sutherland Metropolitan Council, which uses the top floor as a community center. Sightseers are free to roam the building daily for a nominal fee. Visiting Americans were admitted free of charge during the Bicentennial.

At the house, you will encounter those rare American tourists who roam far beyond the London-Oxford-Stratford perimeter searching for something different and more remote. Or you may meet, as I did, a British clergyman who had spent some time preaching in West Virginia, and becoming interested in George Washinton and Mount Vernon, made a point on his return to Britain of visiting Washington Old Hall.

Washington Old Hall is an impressive sight. Its high stone walls, although not quite matching a medieval fortress-castle, give the impression of having been rather foreboding to medieval marauders.

President Carter and other American travelers of today will not see the house exactly as it was when it became the home of George Washington's ancestor, William de Hertburn of County Durham, in 1183. (William took from the estate the name of de Wessyngton, which became Washington). After the sale of the house by William's descendants in the 17th century, it was partially demolished and then rebuilt on the old foundation. But it is believed that all or most of the walls of the kicthen wing are survivors of the original building.

A double archway connecting the flagstone-paved kitchen and the Great Hall, or main ground floor room, is a restoration of the way the house looked in the 1100s. The Great Hall is where the family and retainers would have taken their meals, with the head of the house on a dais at one end of the room. A drawing room on the opposite end of the Great Hall is decorated with 17th-century paneling brought from another area house to illustrate the work of the period.

Along the stairway to the second floor is a genealogical chart purporting to trace the "Descent of George Washington from King John and 25 Barons Sureties of Magna Carta." A recent photograph shows Princess Anne signing the register in the house in 1974.

The line of descent to America's George Washington follows through the great-great-grandson of the original William, John Washington, who, when married, moved to his wife's estate across country in Westmoreland County. Nearly three centuries later, a direct descendant, Lawrence Washington, having made a fortune in the wool trade, turned up further south.

Here Sulgrave Manor, a better known Washington family ancestral home, enters the picture. Sulgrave is only 70 miles from London. Day-trip bus runs are available from London and so Sulgrave Manor, as a Washington family root center, is known to more Americans than the more distant Washington Old Hall.

Lawrence Washington purchased the land of Sulgrave Manor in 1539, following Henry VIII's dissolution of the monasteries, and built his house on the site. He commuted to the city of Northampton, where he served as mayor. It was from Sulgrave Manor that the more immediate ancestors of America's first President stemmed.

Borned here was another Lawrence Washington, a grandson of Sulgrave's purchaser, who became a clergyman with a degree from Brasenose College, Oxford. The Rev. Washington took up residence elsewhere and his son, Col. John Washington, emigrated to Virginia in 1656. John Washington was the great-grandfather of George Washington.

In this connection it is interesting to note that genealogical records in Georgia indicate that President Carter's forebears came to America from England even before George Washington's.

Jimmy Carter represents the seventh generation of his family to live and work in Georgia, beginning with the arrival of Kindred Carter from North Carolina in the late 1780s. Kindred was of the fifth generation in descent from a Thomas Carter Sr., who settled in Virginia in 1637, nearly 20 years before John Washington.

Sulgrave Manor, like Washington Old Hall, is sizeably by current standards but is described in England as a "small manor house." Built of local limestone with a sloping tile roof and well-kept gardens, it has a charm all its own. Washington family members occupied the house until 1659.

As also happened with Washington Old Hall, Sulgrave Manor had fallen into poor condition when, in 1914, it was decided to restore the site as a fitting monument to 100 years of peace between Britain and America. Both British and American contributions went into the project and the house had been open to the public since 1921.