Dickens's thinking comes across to readers along with his vivid depictions of London's lower-class neighborhoods.

Literal

Dickens [gen-の] thoughts [subj-が], London [gen-の] downtown [gen-の] vividly-done depiction [with-と共に] reader [to-には] come-through.

~と共に 'together with / along with' is a formal written pattern. 下町 'lower town' is a cultural-specific term for the older, working-class neighborhoods of Edo/Tokyo, with social and aesthetic connotations beyond a bare translation. 生き生きした 'lively / vivid' is a reduplicated mimetic used as a prenominal modifier via した. 伝わる 'to be conveyed / to come through' is the intransitive counterpart of 伝える 'to convey' — the speaker's ideas reach the listener without needing an explicit agent.