Quietly and without fanfare, an era is drawing to a close in Washington's banking community.At the end of the year, the United Mine Workers of America formally will transfer its controlling interest in the National Bank of Washington, the oldest and third-largest in the city, to a team of investors that includes senior management, New York investors, Washington businessmen and a Saudi Arabian financier.

The previously announced sale got the final go-ahead last Friday when, bank officials said, federal regulators gave word that they had approved the plan. The conclusion of the deal will end more than 30 years of union ownership of the bank, a tumultuous period in which the union's pride and emotional commitment to the bank eventually were overcome by an inability to realize an adequate income on the investment.

In the short term, the changing of the guard scarcely may be felt outside NBW's boardroom. Some alterations in the composition of the bank's directors will accompany the transaction; otherwise, no significant changes in operations are expected, as the current management is set to stay in charge.

But bank executives and observers say that, over the long haul, the sale will help NBW position itself to take advantage of the changes being wrought in the Washington-area banking industry by the rise of interstate banking, deregulation and rapid economic development.

The new investors bring a deeper source of capital for the bank, as well as an array of national and international contacts that they say will allow the bank to attract new business in an increasingly competitive marketplace. "I have long been thinking that the bank -- for capital reasons, for modern financial reasons -- needed to be in different hands," said Luther H. Hodges Jr., chairman of the bank. "By finding the ability to put in more capital, you are able to compete better. You have to grow, with all these things happening in the business."

Officials in the banking industry describe Hodges, a former deputy secretary at the Department of Commerce in the Carter administration, as the moving force behind the sale. Hodges took control at NBW in 1980 after federal regulators forced the bank to hire new management in the wake of a scandal over insider loans. Hodges said he has long been looking for an opportunity to sell.

Perhaps the biggest problem was convincing the union to go along. Under the direction of its legendary founder John L. Lewis, the Mine Workers union bought a controlling interest in the bank in 1949, a bold move in those days. As with other unions that went into the banking business, ownership was more than simply a point of pride: The union needed a ready source of funding that it could not always obtain from other banks.

"It is clear that the bank was, during part of the 1950s, a useful tool to the union in organizing new members throughout the United States and Canada," said Michael H. Holland, general counsel for the Mine Workers. Subsequent changes in banking and labor laws, Holland added, "really didn't provide for those same types of opportunities in 1985."

As time wore on, moreover, the realities of modern banking -- as well, some say, as the union's inexperience in running a bank -- caught up with the Mine Workers. NBW funds were used at various times by union executives for their own internal political purposes, and matters hit rock bottom in 1980 over a series of loan scandals.

Newspaper headlines touted the bank's preferential treatment of borrowers close to Mine Worker officials. Many of these loans went sour, costing the bank millions of dollars, and federal regulators eventually forced the union to give up all influence over day-to-day management of NBW's affairs.

Under close scrutiny by regulators, NBW has since cleaned up its act and regained profitability, but for the Mine Workers the investment increasingly did not make much sense.

The union was told by its investment advisers that if it took its money elsewhere it could make at least seven times the approximately $1 million in dividends that the bank has been providing recently, according to Holland. It certainly didn't help matters that the bank's dividend growth was restricted, despite its improvement in earnings, as a result of NBW's agreement with the Office of the Comptroller of the Currency, the federal agency that regulates nationally chartered banks.

By 1984 the union decided to buck its tradition, and put its more than 75 percent of stock in the bank's holding company, Washington Bancorp., on the auction block.

Holland said that the union also gave its investment adviser, Goldman, Sachs & Co., instructions not to sell to any anti-union group, an important stricture given that the bank itself has an organized work force.

After entertaining a number of offers, the union decided last spring to accept a $70 million bid from a team of investors headed by Peter Del Col and Roy B. Simpson, a pair of New York financiers. The team put $25 million up front and began to arrange the rest of the financing, as federal regulators sifted through the deal.

The Federal Reserve Board agreed to the transaction last month, and on Friday bank officials said they received word that the Comptroller's office would lift the restrictions placed on the bank in 1980. That was the final hurdle for the sale, and officials said they would close the deal by the end of the year. New Owners a Varied Lot

The bank's team of new owners is a varied lot, who appear to have been drawn together not so much by design as by luck and Hodges' numerous contacts in the financial world.

Del Col and Simpson, the leaders of the group, previously were associated with Bradford National Corp., a financial services company that had rocketed to profitability in the 1960s and '70s by providing critical record-keeping services for banks and other financial institutions. The company's fortunes soured in the early '80s, however. Del Col and Simpson were forced out in 1983 after a proxy fight and, according to an associate, "were kicking around for another financial company to be involved with."

They found one in the National Bank of Washington, which Hodges, a longtime acquaintance, brought to their attention. Hodges said he has known Simpson for more than 20 years, since their days in North Carolina, where Hodges served as chairman of the North Carolina National Bank and Simpson was in the textile business. Hodges stresses that he didn't "take sides in the process," but once the deal was signed he joined the new team as an investor in his own right.

Hodges said, however, that he did put Del Col and Simpson, as well as other potential buyers, in touch with Edward L. Palmer, a prominent banker who formerly chaired the executive committee of Citibank. Palmer is on the board of the Saudi Investment and Finance Corp., an international investment firm. As Hodges tells it, Palmer told him of the desire of two executives associated with the company to buy 25 percent to 30 percent of a bank in the United States.

The two investors are Ziad H. Idilby, a U.S. citizen who once served as an executive with the First National Bank of Chicago, and Wafic R. Said, a Saudi Arabian merchant. The two could not be reached last week in their London offices. Idilby and Said eventually will own, through a company set up for the sole purpose of investing in the bank, NBW's largest block of shares: approximately 25 percent, according to Hodges. With their participation clinched, Hodges, Del Col and Simpson proceeded to put together the rest of the financing.

"Our goal was not to have people with$10 million investments. Our goal was to have people with $1 million," said Douglas W. Hawes, a New York attorney associated with the Del Col/Simpson team.

More than 50 investors have signed on, including C. James Nelson, president of the bank; Robert B, Washington, a Washington attorney; Edmund B. Cronin Jr., president of Smithy Braedon Co., and Brent Scowcroft, former national security adviser and now president of Kissinger Associates.

The investors have raised more than $90 million, far in excess of the amount they agreed to pay the Mine Workers, according to officials with the group. The excess funds, they say, have gone in part to buy up some more outstanding stock, as well as to plow an extra $8 million in capital into the bank.

After the deal is closed, however, the investors will not be involved in day-to-day management, although many of them will be members of the bank's board of directors. Their leaders stress that they have no experience or interest in running a bank.

"This is not a group of people that's taking over the bank," said Del Col. "Luther and his people have done an excellent job of improving the earnings . . . . You won't see a tremendous change in the nature of the bank. This is not a turnaround."

This is not to say the investors won't be paying special attention to the bottom line. They paid a bargain price for the bank -- a cheap price indeed, according to some analysts, who say that the investors could well be in a position in three to five years to sell the bank to an outside institution for a tidy profit.

The sale price for the bank was about $66 per share, or 1.3 times the bank's book value last spring. Patrick C. Ryan, senior vice president at the brokerage house of Johnston, Lemon & Co., said that this price is comparatively low compared with other bank transactions recently undertaken in the area.

Ryan cited as an example the recent acquisition of Washington's NS&T Bankshares Inc. by Richmond's United Virginia Bankshares Inc., for which the sale price was 2.5 times the bank's book value.

"I'm not suggesting that they're buying the bank to sell it," said Ryan. But he noted that the bank has found itself in an enviable position: "Anything that should happen in so far as merger is concerned, or the perception of merger, is gravy." John J. Mason, NS&T's chairman, added, "They bought it at a very attractive price, and I don't think they could have bought it for that today."

For his part, Hodges doesn't deny that a future sale is possible, but said that the main interest of the investors is to "build a bank of high quality and headquartered in Washington, D.C." He also pointed out that the banking industry's fortunes, with the Supreme Court's recent ruling allowing interstate banking, look better than they did when the sale first was being discussed.

Hodges, who has emerged as a leading spokesman for the Washington business community, added: "It is inevitable that the face of banking will continue to change. This bank is not going to be absorbed into a Richmond or Baltimore bank, like the others. Our vision of Washington is larger and maybe bigger."

Hodges and the rest of NBW's management team have their hands full. The bank was stung in recent years by several bad domestic loans -- including more than $9 million to Columbia Data Products, a failed computer company -- and some overexposure in international lending, analysts say. Earnings Comparatively Strong

Setting up reserves to cover the bad loans have depressed the bank's earnings, the analysts point out, although NBW, with $1.4 billion in assets, continues to do fairly well compared with its larger competitors, Riggs National Corp. and American Security Corp.

Gilbert W. Keech Jr., an analyst at Johnston, Lemon, said that Washington Bancorp. showed a 0.72 percent return on assets for the first nine months of 1985, compared with 0.66 percent for American Security and 0.64 percent for Riggs. The return on equity was 12.55 percent, above American Security's 10.73 percent and below Riggs' 13.41 percent.

Despite these figures, Keech said that "there's significant improvement to be made." Net income through the first nine months of the year totaled $7 million ($5.06 per share), compared with $7.9 million ($5.77) in the same period in 1984 -- a result, he pointed out, of the reserves set up to compensate for the bad loans. NBW's year-end income is expected to exceed last year's $8.2 million ($5.93), although not by as much as had been anticipated, Keech said.

"I think it's basically behind us," Hodges said of the increase in problem loans. He said that the bank will be much more careful in the future with respect to loans of the magnitude of the Columbia Data deal.

"We are, unlike a lot of banks in the area, an aggressive lending bank to the business community," he added. "We're going to make mistakes. But we're not going to make bad mistakes as we did in the past."