Does British and American English pronounce km differently? Yes, but this is more of a global difference, and not specifically between British and American forms. In some places they will say killohhh-mitter, and other places keelomeeter. Unfortunately my keyboard doesn't have the phonetic alphabet font, so I hope you can understand my phonetic spelling.

We put the accent on the second syllable, "kilOMeter." BUT we rarely say it, except when traveling in a foreign country. In real life, I've heard "klick" more often than "kilometer." "Klick" is military slang for "kilometer." For a short period of time, just a few years, during the 1970s, it appeared as if we were going to convert to "the metric system" (as it is called here). Road signs began to give distances in both miles and kilometers. But the effort was abandoned. We measure distances in miles (and the rest of the "U.S. customary" system--yards, feet, inches--the height of a horse is measured in "hands"). I don't know whether chains and furlongs are still anywhere, but it wouldn't surprise me. A five kilometer race is described as either "a five thousand meter run" or "a 5K run" (read as "five kay").

Yes, Americans stress the second syllable, British the first.