What struck me was that my Canadian friend pronounced it ˈpuːmə, whereas I say ˈpjuːmə.
According to the OED the immediate source of this word is Spanish (and ultimately Quechua). In the case of Spanish and Italian words containing stressed u, anglicization always involves something of a contest between a foreign-style uː with no preceding palatal semivowel and a more native-style juː with one.
In the case of the island of Cuba the ordinary English pronunciation is ˈkjuːbə. It would be most unusual to pronounce it Spanish-style with no j.
Canned/tinned tuna likewise seems to be pronounced English-style. We Brits, who have tj- or its successor tʃ- in tube, tutor, tune, say ˈtjuːnə, ˈtʃuːnə (making it for most of us a homophone of tuner). Those Americans who have plain t in tuːb, ˈtuːt̬ɚ, tuːn say ˈtuːnə. (This is not exactly a Spanish word, since in Spanish the fish is atún, from Arabic تون tun and Latin thunnus, Greek θύννος thynnos. It is not clear why we stopped calling it ‘tunny(fish)’ and started calling it ‘tuna’. The OED says it’s American Spanish.)
In the phrase numero uno (actually of Italian rather than Spanish origin) the nu- part has or doesn’t have j according to the way we pronounce new and nude; but the uno always, I think, has plain uː with no preceding j.
Utah always has juː-, but several other words seem to be variable, with Brits tending to include j and Americans usually omitting it: barracuda, iguana, jaguar,
Back to puma: I hastened to check whether I had included the yodless form in LPD. I was relieved to find that — unlike the OED and the ODP — I had. The fact that it is in thin (not thick) black type after the || shows that it is what the database calls an AME VARPRON, i.e. not the main AmE pron. I continue to regard ˈpjuːmə as the MAINPRON for both BrE and AmE, whatever my Canadian friend says. Perhaps I ouɡht to do a preference survey.