When saying a private portion of the service, such as pesukei de-zimra, may the shaliach tzibur (aka chazzan, or the person leading services) use English, and switch to Hebrew for the portions said aloud?
Bais Yosef rules that one may pray in any language.
Mishnah Berurah seems to suggest that one may either pray in Hebrew (whether or not he understands the words) or in any other language (but ONLY if he understands the words).
At a time when the Sha"tz was actually there to recite the prayer on behalf of the others, it might pose a problem to say most of the text in one language, and then the last line of every paragraph in Hebrew; the complete text is not being recited in either language, as one whole unit, which would seem to pose problems in fulfilling ones (or another's) obligation to pray.
Generally speaking, we do not rely on the Sha"tz to exempt others from their prayer obligations in our times. Nowadays, the Sha"tz is more symbolic than functionary. From that point of view, the Sha"tz technically doesn't have to say anything at all during the silent parts.
Based on that understanding of the Sha"tz's function in contemporary synagogues, one may indeed recite the silent parts in English, and the vocal parts in Hebrew.
However - I would caution that for kriyas shema, and for the silent shemoneh esreh, when typically the Sha"tz himself intends to fulfill his own mitzvah of kriyas shema and of tefillah, that he pick one language for the entire section and stick to it.
(Of course, one could recite all of kriyas shema in English, silently. Once he is finished with that silent recitation, he could switch to Hebrew for the vocal "Hashem Elokechem Emes" - and it wouldn't violate the concept of saying the entire section in one language, because the entire kriyas shema was already said silently before the Sha"tz recites those words out loud, to alert the congregation to move on.)