Diacritics are usually stripped in air travel. In Hungarian we have many letters with diacritics, but it is never a problem that the passport has them and the system doesn't.
Not in all cases. In Germany and Finland (maybe all EU passports???) ä is spelled ae, ö is spelled oe in the machine readable part (umlauts shown in the "human-readable" part). This is important to know when you need a visa.
For Germans this is not a big problem because it has been like this forever if the umlaut is not available for technical reasons. For Finns this is a problem, because this "transcription" is completely unknown in Finnish. For a couple of weeks now it has been possible to get an electronic visa for Russia on the internet. Reportedly many Finns with an ä in their name (that's not uncommon) dropped the dots when applying for their visa, because an ä is not accepted. At the border they were not allowed to enter, because the machine-readable part of the passport has ae instead.
There is an ICAO recommendation. However, it is not unambiguous and of course it's not legally binding. So in the end every country decides what they do. (Possibly there are more multinational agreements e. g. inside EU, but I doubt there is anything truly worldwide.)
For German names, this is a problem. I have an ü in my name and this is transcribed as a "ue" in my passport. Transcribing it as u would produce a different name (which AFAIK actually exists).
In Hungarian the diacritics are also important, for example Szilasi and Szilási are different and are pronounced differently. Still, it won't be an issue when flying or other stuff.
German is more complicated though with all the substitution rules.
Not to mention Germans who actually have an ue in their name, still pronounced as ü, but written as ue only, never as ü. Or someone may be called Gross, but it would be incorrect to write it as Groß, while someone else's name may be Groß with the acceptable alternative spelling Gross when ß is unavailable.
