Learn Italian: Purtroppo non c'è nessuno scampo per loro. - Unfortunately there is no escape for them.

Because the noun scampo starts with an "impure s" (an s followed by another consonant, which is c in this case). Just like you use the definite article lo instead of il for these words (lo scampo), you must use the full form nessuno instead of the shortened nessun.

Why do we use non here if nessuno already means "no"?

Italian requires a double negative when negative words like nessuno, niente, or mai come after the verb. Since nessuno is placed after the verb c'è, we have to put non before the verb. If nessuno was at the very beginning of the sentence, the non would be dropped.

What exactly does scampo mean? Is it related to the shrimp dish "scampi"?

Yes, they are related! Scampo means "escape" or "salvation" (from the verb scampare, meaning to escape). The seafood scampi got its name from the rapid way those little prawns flick their tails to escape danger. In this sentence, nessuno scampo is a common expression for "no way out".

Could the sentence work without nessuno, just saying non c'è scampo?