If you've swallowed something dangerous, what you do about it depends on what you swallowed.

Literal

Dangerous-thing [object-を] swallowed case, swallowed thing [depending on-によって] coping-method [subject-が] differs.

Emergency-first-aid style instruction. ~た場合 ('in the case that X has happened') is a formal conditional frame, used especially in instruction manuals, legal text, and official guides. 飲み込む is a compound verb: 飲む ('drink') + 込む ('into') = 'swallow (food/object).' ~によって here is the causal/dependent use — '(differs) depending on X.' 対処法 ('coping method,' 'how to respond') is a Sino-Japanese compound — 対処 (cope) + 法 (method).