Dejeuner: The breakfast of an American who has been to Paris. Variously pronounced.
Being a translator and a Quebecer who’s travelling in Spain, this is a matter of some interest to me. Basically, I try to go by whatever the prevailing English-language usage is, except sometimes.
There isn’t one blanket rule. For example, in general, smaller places get a native-ish pronunciation (gehr-NEE-ka, not gurr-nika, for Gernika - btw, I’m using the Basque spelling as it’s an important place in Basque culture, and also because it neatly distinguishes it from the painting; in fact a mural of Guernica there called the “Guernica Gernikako”).
Larger places, it depends. In a lot of places, the name is spelt the same, diacritics aside, and it’s pretty uncontroversial to pronounce it Englishly: Madrid, Barcelona, Valencia, Granada, Montreal, Paris, and so forth.
In other places, there was a historical English name that’s now all but taken over by the native name: Córdoba, Zaragoza, Vizcaya (or Bizkaia), and such. This could be recent, such as Beijing, which seemed to complete its transmogrification at around the time of the Tiananmen Square massacre; now “Beijing” is pretty much uncontroversial and I haven’t seen anyone write “Peking” in anything but historical references in the last fifteen years.
In other places, the native name is coming up fast: Mumbai, for example, or Côte d’Ivoire. Still other places are a bit further behind: I doubt many people in the English-speaking world (apart from India) have heard of Kolkata or the like.
Finally, there’s the case where there’s a consistently used, different, English-language name: Lisbon, for example, or Rome. There’s just no point in saying “Roma” or “Lisboa” in English besides appearing to be a snot (or the editor of an edgy travel guide). It’s not controversial and everyone says it, just like for names of countries (it’s Greece, not the Elliniki Thimokratia).
The important thing is to keep abreast of usage. Who knows? Maybe in forty years we’ll all be saying Lisboa. But I doubt it.
Of course, the stupidest thing of all is to try to pronounce an English name natively: saying mü-nihhh for Munich, for example.
A side note: in writing about Quebec place names, especially smaller features such as streets, I follow the official rule that the French is used in both languages: so I write my address as being on rue Saint-Quelqu’un, not Saint Quelqu’un Street, even if the latter is what I would say out loud. In way too many cases the English name is just confusing, for example when Hamish, recently arrived in Montreal, was directed to an address on “Pine Avenue” and wandered around for hours before he realized he ought to be looking for av. des Pins. And Og help you if someone told you to go to “Dorchester Boulevard”.