I'm not a historian but a Slavist with some training in Slavic historical linguistics.

The Old East Slavic word is actually spelled Московь: [moskov'], which is transliterated "-ow" in West Slavic and Germanic languages. We can see this -O- spelling in words like "Muscovy" (the term for the medieval principality - Московия in Contemporary Standard Russian) and "Muscovite" (one who is from Moscow - Москвич in CSR). The [ov] ending appears in the earliest text of what would become Russian, the 12th century Lavrentian Primary Chronicle.

In the late Middle Ages, though, vowels began to mutate, especially towards the end of words. This was partially due to what is called the Jer shift. Jers were mini vowels that ended almost all masculine nouns and often existed within what are consonant clusters CSR (such as the -skv- in Moskva). As these demi-vowels stopped being pronounced, the language accommodated, putting one full vowel for every two jers (roughly). This is not necessarily what is going on in "Moscow," but it shows that the East Slavic language was very much in flux.

I believe that what we actually see in the switch from "Moskov" to "Moskva" is a "fleeting vowel," meaning presence of the -o- varies across cases and derivations:

• Московь (Moskov')
• на Москви (na Moskvi - in Moscow) - As an aside, "na" is a locative preposition used for bodies of water (among other less relevant things), which implies that the name for the river Moskva preceded the name of the city.

The Russian spelling Москва/Moskva begins to appear in the 14th Century, well after trade had been established with western nations.

Source:

Max Vasmer's Etymological Dictionary (in Russian): https://dic.academic.ru/contents.nsf/vasmer/

Edit: Clarity/typo

Москва is derived from Москов- (the river), in turn from Москы. As Moscow was a significant city quite early - many buildings of the Kremlin, for example, was constructed by Italian architects in the late 15th century - there was much greater interaction with foreigners, and Московь was reasonably glossed as Moskau, Moscou, Mosca, and Moscow.

Киев (Київ) did not see any such transition within Russian, and the original was preserved (but you will find historic references to "Kieff"). There are certainly other cities that have lost the original pronunciation - Warszawa becomes Warsaw or Warschau, Łódź becomes Lodz, Szczecin becomes Stettin, Ростов-на-Дону becomes Rostov-on-Don, Архангельск is Archangel/sk, and so on. Санкт-Петербург, with a very large foreign population and international significance, is Saint Petersburg.

It's also worth noting that other major Russian cities are generally quite a bit younger than Moscow, with a few exceptions: Екатеринбург and Новосибирск were founded in the 18th and 19th century, for example, while other cities had less significant Western relationships that would enduringly influence foreign pronunciation.