If an English diplomat spoke to an Italian diplomat in Italian, then I would expect to hear them both talk about "Londra" rather than "London". However, if they were talking together in English, I would expect them both to say "London".
If a French diplomat speaks to an Italian diplomat in English, as your question allows, then surely they should both talk about "Michael-angelo" rather than "Michel-Ange" or "Michelangelo."
It is the diplomatic language being spoken that dictates the pronunciation of the name - otherwise confusion reigns,
When it comes to names of noteworthy people, we should follow the same procedure as above, provided that the language already has a well-established pronunciation.
For example, both parties should say "Christopher Columbus" when speaking in English, "Christophe Colomb" when speaking in French and of course "Cristoforo Colombo" when speaking Italian. There is a reason for this. There may be others listening or taking part in the conversation. They could be of any nation. If they all speak 'diplomatic' English then that is the language they understand. So, the Chinese ambassador would understand Columbus but perhaps not Colomb.
The same rule should apply for other names. However when the pronunciation is very similar then I don't think it would be a faux-pas for the Italian diplomat, for example, to pronounce 'Michelangelo' in the Italian way when speaking in English. Both parties would understand and that is what is important. However if the Italian speaker insisted on using the Italian pronunciation in an exaggerated way when speaking English, it would surely come across as a criticism and even be rather rude.
When it comes to names that have no accepted equivalent in the language being spoken then I think the diplomat whose language is being spoken should make some attempt to use the pronunciation of the language of origin. If this proves difficult, for example with the African click-languages, the diplomat should make his or her best attempt. Where this could cause most problems would be in respect of a tonal language such as Mandarin. In those tongues, the meaning of a word can be totally changed by its 'melody'. It would pay dividends for a diplomat to take professional advice before attempting some Chinese or some other Eastern Asian names.
Speakers from the USA often don't know how to pronounce British place names. So, for instance, as a Brit, I pronounce "Worcester" as "Wooster". If an American spoke to me about the English city of "Wor-sester", I would excuse it for the first few times but it would become rude and an implied criticism if he or she insisted on doing so for evermore.