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