Douglas H. Ludeman was named chief operating officer of United Virginia Bankshares Inc. last week and, as a result, is likely to become the next chairman of the company that owns Virginia's largest bank.

Joseph A. Jennings, the current chairman of the $5 billion bank company, will retire when he turns 65 in Sept. 1985.

Ludeman, 52, already is president of the Richmond-based parent company and its United Virginia Bank and a director of both. He will add the new title of chief operating officer on Jan. 1, the bank said.

The bank's board of directors also named Richard G. Tilghman as vice chairman of both the bank and the holding company. He will report to Ludeman and take over many of the activities that Ludeman now oversees, including regional Virginia banking. Tilghman, 43, is executive vice president for corporate banking.

United Virginia has 191 offices throughout the state. It will lose its status as the state's biggest bank company if the proposed merger of the state's second-biggest and fifth-biggest bank companies--Virginia National Bankshares Inc. of Norfolk and First and Merchants Corp. of Richmond--is finalized. United Virginia and Dominion Bankshares Corp., the state's fourth largest, had talked merger last spring, but Dominion broke off the talks.