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The president's speech went on so long that the second half of the ceremony got rushed.
Literal
Company-president [attributive-の] greeting [subject-が] long-[te-reason-くて] ceremony [attributive-の] second-half [subject-が] push-push [to-に] became-[unfortunately-てしまった].
The speaker complains about a speech running over. 挨拶 here means the formal opening/closing remarks (not everyday 'hello'). 押せ押せ here means 'rushed, behind schedule' (from the feeling of being pressured/pushed from behind) — note how this differs from 押せ押せムード in sports (attacking momentum). Same word, two almost-opposite contextual meanings. ~てしまう marks the regrettable outcome.