Barnsley: Weather Forecast. The warmest day over the next 7 days in Barnsley is forecast to be 20th April at 15°C (59°F) and the warmest night on 20th April at 7°C (45°F). The average temperature over the next 7 days is forecast to be 12°C (54°F) and there will be 5 days of sunshine . The average for April is 0°C (0°F) in Barnsley.