The affricate /t͡s/ is not considered to be one of the basic sounds of English (it's not an English "phoneme"). English speakers do undeniably make use of a consonant cluster /ts/: you can find it at the end of plural nouns like bats or crates or third-person singular verbs like hits or fights. It just isn't considered to be a sound of its own: it's just a combination of the /t/ sound and the /s/ sound.
The consonant cluster /ts/ doesn't occur at the start of any native English word. It's not particularly difficult for an English speaker to pronounce /ts/ in this position, but I'd compare it to other consonant clusters that don't occur at the start of native English words, like /ps/ or /kʃ/.
Foreign words that are spelled with the letter Z may be pronounced by English speakers with /z/, even if the original language had another sound. This is the case for zucchini (pronounced /zuˈkini/ in American English) and influenza. In some words like zeitgeist, you might hear either /z/ or /ts/. Rarely, this kind of variation leads to confusion in the other direction: the word chorizo, where Z represents /θ/ or /s/ in Spanish, is apparently pronounced as "choritso" by some English speakers.
Proper names should be pronounced as they are by the person so named, or the locals of the place so named. However, some languages don't normally use some sounds, and a close approximation may have to suffice. For instance, Tsar is often pronouncd Zarr in AmE. – Davo – 2017-06-19T11:31:11.953