You can apply for an Estonian drivers licence if you have an
Estonian ID code / residence.
An Estonian driving licence can be applied for by a person who
lives in Estonia for at least 185 days during the calendar year or
who can prove that they have studied in Estonia for at least six
months and whose right to drive has not been suspended, withdrawn
or revoked in the state that issued the driving licence or
Estonia. Permanent residence is proven by the data of the
population register. The person is required to prove that they
permanently reside in Estonia.
Check from here to
see exactly what you need to do in order to change your licence to
Estonian driving licence.