Compatibility summary
You will need a travel adapter in South Korea; plug types used there differ from United Kingdom.
Plug comparison
United Kingdom plug types: G
South Korea plug types: C F
There is no common plug type between United Kingdom (G) and South Korea (C F). You need a travel adapter.
Voltage comparison
United Kingdom: 230V, 50Hz
South Korea: 220V, 60Hz
Voltage difference is within a range many devices tolerate; check your appliance label.