If you catch a cold, it takes a long time to shake it off.

Literal

Cold [object-を] catch-[if-と], cold [subject-が] stubbornly doesn't-leave [emphasis-よ].

風邪がなかなか抜けません means 'the cold stubbornly won't go away.' なかなか with a negative means 'not easily/stubbornly refuses to.' 抜ける means 'to come out/leave.'