Green’s Dictionary of Slang

1. (US Und.) a small-town police officer; a private detective; a badly dressed police officer.

1950		Goldin et al. DAUL 19/2: April fool copper. A private detective; a small town, shabbily uniformed policeman.
1952		R.S. Prather Darling, It’s Death 5: Ain’t you that lousy April-fool copper from LA?
1985		R.S. Prather Have Gat Will Travel 10: A slewfoot. An April-fool copper.

2. (US prison) a prison guard.

1992

Bentley & Corbett Prison Sl. 96: Dick […] any policeman or prison guard. (Archaic: April fool copper).