Libra run a service between Frome and Midsomer Norton which passes through Great Elm every 2 hours, but unfortunately, not at weekends.
See Libra Timetable
Trains run from
Frome (3 miles) – for Westbury and Weymouth
Westbury (10 miles) – for Warminster and onto Salisbury, also to London.
Trowbridge (10miles) – for Bristol.
Bath spa (14miles) – for Bristoland London
See Trainline for train times
Nearest airport is Bristol (28 miles)
For flights see Airport website