I am not familiar with any academic sources that indicate there was any official renaming of the city after the conquest; likewise, it's not correct to say the city was renamed in the 1920s or 1930s, since the name in use by the government went unchanged.

Instead there were and are different names used in different languages, and with a few exceptions Istanbul was simply the name used in Turkish before the conquest, after the conquest, during the transition to a republic and through to the present day. Speakers of other languages use other names.

In the late empire (my area of study), the most common form was Istanbul, which you see in official government correspondence, maps, newspapers and, my favourite, the yearbooks that were issued every year for each province or for other state institutions (ministries, the military, etc.). As an example, this salnâme (yearbook) shows the place of publication as Istanbul (if you can read the Arabic alphabet, it's visible on the second-to-bottom line on the second page of the PDF). This yearbook, on the other hand, uses Dersaâdet, another commonly-used name in official documents and newspapers.

This isn't to say that Constantinople was never used; there were a few coins that were minted using the Arabic/Turkish form of the city. However, these are absolutely some of the few exceptions to the general rule, and the more common use of Constantinople in any official documents was in French or other foreign languages. Using another yearbook as an example, this one shows how in Ottoman Turkish the place of publication was Istanbul but in the French portions it was Constantinople.

I think this is where the confusion in sources such as Wikipedia comes from: the city was not renamed in the 1930s. Instead, it was simply that the Turkish postal service decided to no longer accept mail addressed to anything except Istanbul, leading to the most commonly-used name changing in languages such as English, but not necessarily in all languages.

So in summary inasmuch as control over the city dictated its "official name", 1453 is when it is correct to say that it became Istanbul, but not because of some official decree but simply because that was the name most commonly used by the Turks that conquered it.

Edit: I forgot to mention that the name Istanbul predates the Ottoman conquest of the city and was recorded in use by Ibn Battuta, the famous Arab traveller, a century before.

Edit 2: Added summary paragraph.

Strictly speaking, it's not true that Mehmed II renamed Constantinople following the conquest, although you will occasionally find this notion repeated in certain secondary sources. The Ottomans continued to use the name Constantinople (in its Arabic form, Kostantiniye) in a variety of official functions, such as on the empire's coinage and in official documents produced in the capital. Consequently one occasionally reads that Constantinople was the "official" and Istanbul was the "popular" name, but this is also misleading. Certain official titles in the 16th-18th centuries in fact utilized the latter. For instance, the chief judge who presided over area within the city's walls was officially referred to as the judge of Istanbul - not Constantinople. Thus it seems that, at least during the early modern period, Istanbul could refer specifically to the walled city (today's Fatih district) whereas Constantinople was used to refer to the larger urban area including Galata and Üsküdar on the Asian shore. The two names therefore existed side by side, along with a third said to have been invented by Mehmed II himself: Islambol, a pun on Istanbul meaning "full of Islam" that became a kind of pet-name for the city among certain writers in the ensuing centuries. The frequency of use and exact connotation of these names may have changed during the nineteenth century, and given that I have much less familiarity with the conventions of that period, I can't comment on them.