Here's what Michael Quinion has to say on the subject:
[Q] From William Murray; a related question came from Tim Riley in London:
"The title of your new book, Ologies and Isms, reminded me of a thought I've
had about kilometer. The word itself is not essential in the USA, except for
scientific measurements. I use two different pronunciations depending on
context: in scientific and most contexts, I'll use KILometers, but for road
distances (and common measure when I'm in Canada), I'll say kiLOMeters. I
wonder if this dual pronunciation nicety is common in the rest of the
English-speaking world."

[A] The version with the stress on the second syllable used to be rather
more common in the US, with the rest of the English-speaking world tending
to stress the word on the first syllable. In Britain and the Commonwealth
the one stressing the first syllable is still considered by some to be the
only correct form.

But the situation is changing. I hear it stressed on the second syllable
quite regularly from reporters on the BBC news, for example. Professor John
Wells of London University surveyed British usage for the 1998 edition of
his Longman Pronunciation Dictionary and found that in the decade since his
last survey the form with initial stress had lost ground to the other. In
1988 kilometre with initial stress was just in the majority, but in ten
years had slipped back to only 43% of those who replied. If that trend
continues, it will take only a generation for the non-specialist American
pronunciation to take over entirely in Britain. It's ironic that the
American pronunciation should prevail when Americans-almost the last users
of non-metric measures in the world-have less need for the word than other
English speakers.

As you say, in scientific use the stress tends to be more often on the first
syllable, perhaps because the significance of the prefix is more obvious in
technical contexts.

What puzzles people about this stress shift with kilometre is that it doesn'
t occur with most other metric units in -metre (-meter if you're American):
we don't move it in centimetre or millimetre, for example. Also, it doesn't
happen with other units prefixed with kilo-, such as kilogram and kilolitre
(the first of these is unlikely because it has too few syllables, but the
second ought to be possible).

It does happen sometimes with micrometre, however, which can take on the
same stress as the measuring instrument, which is spelled micrometer
everywhere. This gives us the clue to what is happening-it's by analogy.
Every measuring instrument whose name ends in -ometer has the stress on the
antepenultimate syllable, the one containing the o: thermometer, barometer,
chronometer, hygrometer, speedometer, galvanometer, sphygmomanometer, and so
on. The stress on kilometre is merely following the trend. The American
spelling as kilometer has possibly reinforced the connection.

This, of course, only pushes the problem back one step. Why should all these
measuring instruments have their names said this way? It seems in part to be
linked to the concept of strong and weak vowels in English. Some, such as i
and u, are usually weak, while the remainder, including o, are usually
strong. All that these terms mean in practice is that syllables containing
strong vowels (such as the o in -ometer) are more likely to receive the
stress. There is also a tendency in compounds for the stress to move towards
the end of the word.

But these are only tentative suggestions, since the history of language is
full of unpredictable shifts in stress. Professor Wells pointed out that
various historical studies "show that many words have fluctuated wildly in
their stress patterns over the years. Why did we change from cha-RAC-ter to
CHAR-acter, or from bal-CO-ny to BAL-cony? I don't think anyone really
 knows".

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