When President Reagan travels abroad, he is, understandably, provided the fullest Secret Service protection money can buy. His 37-year-old son, it turns out, gets much the same treatment.

Reagan's son, Michael, is on a three-week private vacation in Britain, accompanied by his wife, Colleen, his 4-year-old son Cameron -- and a security escort of 19 Secret Service agents.

The itinerary for Michael Reagan, a California businessman, his family and the security detachment has not been released. But a British newspaper, The Daily Mail, reported that their first stop was the Grosvenor Hotel, a luxurious hostelry owned by the Duke of Westminster in the town of Chester. ("Stylish opulence" is the hotel's hallmark, according to one guide book.)

The Reagan family took the James Mason Suite at 93.50 pounds a night (about $170). The 19 agents, plus the command post, were booked more modestly in single rooms costing 35.50 pounds nightly (about $60). What with food and other amenities, the charge per agent, according to a hotel industry source, had to be at least $100 for each of the three nights the party was on hand -- a total of $5,700 for the agents' full stay.

They arrived in Britain last Saturday and will be touring the country until Nov. 15 -- a total of 23 nights. The American Embassy said it has no official contact with Reagan but does provide assistance on security matters to the Secret Service. That means transportation and communications backup. In Washington, a presidential spokesman said the White House press office had no information on Reagan's itinerary.

The embassy was apparently also responsible for hotel reservations. Instead of the Grosvenor, a secretary sent a confirming telex to the humbler Gloster Lodge, which suspected something was wrong because it has only eight rooms, nowhere near enough to accommodate Reagan's entourage. It, however, has a top price of only 24.50 pounds a night -- about $40.

The owner, William Whitelaw, decided that the telex was probably meant for the grander Grosvenor and the mix-up was resolved in plenty of time for Reagan's arrival. After Chester, a town in Cheshire, near Manchester, the family made for the resort town of Stratford-upon-Avon, the birthplace of the playwright William Shakespeare.

Traveling ex-presidents also don't do badly on the security front. Jimmy Carter was on a private visit to London this week, promoting his new book "Keeping Faith" and seeing "friends and advisers" about a new institute on conflict resolution he is starting at Emory University in Atlanta.

Carter had 12 Secret Service agents with him. They were comfortably ensconced with the former president at London's Churchill Hotel where a single (including service and tax) goes for a tidy 79 pounds, about $125.