When did people stop translating their names when moving from one country to another?

First you should not confuse the monarchs with ordinary people. Monarchs usually called by the traditional forms of their names in many countries both then and now.

Regarding spelling of the names of common people, it depends on the language rules, I an totally sure I would change the spelling of my name when moving from Russia to another country.

There is a difference between translation and transliteration. The Чайко́вский example obviously had to be transliterated, and there are different transliteration conventions in different languages.

Interestingly, the cities München, Nürnberg and Köln usually get translated, while Berlin or Hamburg don't. Does the diaeresis have anything to do with it?

At which point in history did the bureaucracy stop people from altering their names when emigrating from a country to another so that they had to keep the spelling in their original language?

The above premise is not true or at least not true for all countries. In the US for example:

• People with mononyms are forced to make up a "last name" even if they don't have one.
• East Asians are forced to reverse their name order to fit the first name first, last name last order.
• If your name has diacritics (especially ones that are unusual in the English language, such as those in Vietnamese), the system will automatically drop them, however important you think they might be.

This can all be occurring within the Latin alphabet. (I have already mentioned Viet Nam, whose script uses the Latin alphabet. Some Indonesians have mononyms and Bahaha Indonesia uses the Latin alphabet. Singapore, which has a variety of populations, uses English and thus the Latin alphabet. People from these countries who emigrate or even simply travel to the US are forced to follow the US naming format.)

I'm not sure about the situation in European countries, but I suspect similar considerations apply.

You are conflating many different things, some of which still happen regularly:

• Transliteration is still necessary, and it happens a lot, often with differences from one country to the next (cf. Влади́мир Пу́тин/Wladimir Putin/Vladimir Putin/Vladimir Poutine)
• Monarchs' names do get translated, at least occasionally, especially by people who care about royalty. Thus, Elizabeth II is Élisabeth II in French. But there are exceptions (I have never heard anything else than “Juan Carlos” but a comment mentioned the fact that it does get transalted in Hungarian).
• Common people can still change names, but usually that happens when you change citizenship, not merely when you travel, which has become much more common.
• The bureaucracy still forces people to change names all right. I know many people who are stuck with a misspelt or misinterpreted name since they applied for a passport or residence permit.
• You did not mention this but it's not uncommon for people (from China, in particular) to pick a first name like Sarah, Michelle, etc. when they move to Europe or the US and to use it consistently instead of their birth name. Something similar also happens in large multicultural countries like Russia (cf. comment).

But remember that spelling in general (i.e. even for common nouns and other words, not only proper nouns) wasn't always standardized in the way it is today. Another related changed is that many people now know how to write and are expected to fill in forms, etc. Two centuries ago, your name was a spoken word you would tell to a clerk, now it's a written word you have to fill in everywhere. This (much broader) process would seem to readily account for the differences you noted.