The climate in South Korea in January is going to be cold and generally harsher than winters in the UK.
January is usually the coldest month in South Korea and there may be several days where temperatures drop below freezing.
Travelling to South Korea in January requires preparation for conditions which may be below freezing. Snowfall may be expected at higher altitudes.
Temperatures in Seoul tend to be a little higher than in the countryside, purely due to the concentration of buildings and people which help to maintain warmth in the urban landscape. January also tends to be quite dry in Seoul.
Temperatures in Jeju Island are on average 5 degrees higher than those in Seoul.