I just started tutoring a lady in English who hails from Mexico City. She knows little English, so I pronounced Mexico like Hispanics do. She surprized me by pronounced it the way Americans do. My guess is that she's trying to be respectful here. It impressed me.
Cerrtainly, if I were speaking Spanish, I would say "MEH-hee-ko." If I were speaking English in Mexico I might or might not do the same. But if I'm speaking English in the USA, even to a native speaker of Spanish or someone who came from Mexico, I don't feel a need to do that; it feels like an affectation.