On videos, I hear native Ukrainians pronounce it "Ki-yeev", but not stressed, so it might sound like Keev. On Phil's initial point, I did hear a Western presenter say Warsaw, then say "Var-sa-va", presumably the native pronunciation. Interestingly, there seem to be US journalists and presenters using "Mosc-oh" and not the Americal "cow". Still, nowhere near native "Muskva".
The Russian pronunciation seems to be taken as an insult by Ukrainians. I also note that placenames ending in "-iw" seem to be pronounced by Ukrainians as a "w" and not as "v". Given the size of the country and its history, I guess the may be marked variations around the country. And for the desire to take the country under Russian domination, we have to (partly) blame the (now named) Kyivan Rus, or at least Russian's view of their origins.
Oh, and I get annoyed when Putin becomes Pyootin. But that's me.
Bill