kettleer, in Europe, including Great Britain, there are laws that require that every gun manufactured be tested for safety by firing with cartridges of greater than normal pressure. This is called proof testing. If the gun passes, it is then stamped with an official mark, called a proof mark. This is done on every gun, even in wartime. For example, the Germans continued to do it right up to the last day of the war.
The various European countries got organized to accept each other's proof marks long ago, but if a gun is imported from a country without proof tests - like the United States - it is always tested and marked by the country of import. In the case of guns imported by the British government, a variety of other marks were stamped on the gun, like the caliber, the proof test pressure, a government property stamp, and so on.
Your gun has none of these marks. Also, although the British did buy a substantial quantity of H&R revolvers, they were not in 38 caliber as one would expect. They were apparently all in 32 S&W Long, for issue to the British police.
There may be some chance that your father was correct about this gun, because they may have been issued as equipment on American built Liberty ships, which may have been crewed by the British Merchant Marine (not Marines like our Marine Corps - just merchant ship seamen). I don't know much about that subject, or whether the Brits would bother to proof a gun in merchant ship's arms locker. Especially if that ship was due to be returned to the US government at the end of the war, which may have been the case with Liberty ships. That's a lot of maybe, though.
Sorry not to be of much help. Your gun looks minty, and there is increasing collector interest in these older H&R revolver. It looks like the plastic grips may be deteriorating. I think these grips were made of an early plastic called cellulose acetate (aka Tenite) and it has an odd tendency to go bad in enclosed spaces, such as the gun's own box. Apparently the volatile chemicals in the plastic gradually produce a gas, and that gas accelerates the plastic's decomposition. Film collectors call this "vinegar syndrome", since cellulose acetate was also used for making positive prints of movies, and tightly sealed film cans exacerbate the problem. (The gas is said to smell like vinegar.)
I don't have any answers to your specific questions about serial numbers. The man here who would know that, Jim Hauff, has been absent for some weeks after having health problems, and we are worried about him.
PS - thanks for putting up excellent pictures of your gun. When Jim returns, he will be glad to seem them and interested in what you have said about it.