If you passed your Full UK Car Licence test before 1st February 2001 you have entitlement to ride a moped up to 50cc (maximum design speed not exceeding 45 km/h) on your current car licence and therefore are not required to undertake Compulsory Basic Training (CBT). However riding a moped of any size is very different to driving a car and it is recommended that you sit a CBT.
If you wish to ride a moped or motorcycle of 51cc or over you are required to undertake a CBT and renew this every 2 years (unless you have passed a full bike test).
The rules surrounding motorcycle licensing can be complicated and we recommend that you refer to www.gov.uk/ride-motorcycle-moped/overview.