A British television documentary drama about the 1977 execution of a Saudi Arabian princess and her lover for adultery has angered the Saudi royal family and jeopardized British-Saudi relations.

Although no formal threats have been made by the Saudis to cut off diplomatic relations or trade, British officials said today they were "very concerned" about the Saudi reaction, which they described as "very serious indeed."

Britons saw the show on the commercial television network last night. British officials revealed today that Saudi diplomats had protested the scheduled presentation during a two-week flurry of exchanges after they had seen a preview.

The protests were so strong that both British Foreign Secretary Lord Carrington and his deputy, Sir Ian Gilmour, sent messages to Saudi Foreign Minister Prince Saud expressing their "deep regret" but explaining that the British government could not stop the film from being shown.

The British ambassador also was rushed back to Saudi Arabia from a vacation in France to try to smooth matters.

The Saudi royal family is reported to be angry about suggestions in the film that the princess' execution was ordered by her grandfather, an elder brother of the Saudi king, as an example to other young princesses. Other causes for their anger were said to be allusions to corruption and to illicit sex among members of the royal family.

The two-hour drama, "Death of a Princess," is a reconstruction by independent British filmmaker Antony Thomas of the case of 19-year-old Saudi princess Misuaal, who was publicly executed by firing squad in Saudi Arabia in November 1977. A widely published photograph of her lover being decapitated by an executioner's sword helped bring the case to worldwide attention at the time.

Thomas researched and made the film, which is to be broadcast later in the United States and several other countries, in Beirut and Cairo over a two-year period. Fictional names and dramatic effects were used, but Thomas said the drama is the "record of a truthful experience" after an exhaustive investigation.

"Personally, I think this film was very offensive to the whole Saudi royal family and to our country," said an official at the Saudi Embassy here today. "We have our own laws and morals, which we keep to ourselves. It is difficult for anyone in England to understand the moral issues of this matter from a Saudi Arabian point of view."

Thomas told reporters in the Netherlands, where he is promoting the film, that once someone decides to make "a film about an investigation . . . it is immoral to say, 'What are the consequences going to be?' You just have to try to tell the truth."

A spokesman for the commercial television network here said it decided to go ahead with last night's broadcast after listening to "representations on behalf of the Saudi Arabian authorities." The broadcast was prefaced by the explanation that "we have been asked to point out that equality for all before the law is regarded as paramount in the Moslem world."

The spokesman said the network "has no knowledge" about an allegation made in the London Evening Standard newspaper today that the Saudis had offered Thomas and the network more than $10 million to stop the broadcast. In the Netherlands, Thomas refused to comment on this report.

Carrington's expression of regret to the Saudis was criticized today by a number of members of Parliament here. The same thing had happened to his predecessor, David Owen, who apologized in 1977 for the British government's characterization then of the execution of the princess and her lover as a regrettable tragedy.

This new controversy comes at a time when British diplomats have been trying to talk the Saudi government out of administering a public flogging of 80 lashes to the wife of a British surgeon working at a hospital in Jeddah.

The couple, Penny and Richard Arnot, were convicted of violating Saudi Arabia's strict alcohol laws after they hosted a party in Jeddah 10 months ago at which a half-dressed British nurse and a Dutch sea captain fell to their deaths from the Arnots' sixth-floor apartment balcony.