While living in the United Kingdom you should apply for a driving license at the UK Driver and Vehicle Licensing
UK driving license will be issued on the basis of your Estonian driving license. If your Estonian driving license is lost, stolen or expired, the Embassy can provide you with a certificate confirming your driving entitlement. Estonian citizens who have registered their place of residency in the UK can apply for the certificate from the Embassy by post. In order to do so please send us a written request for the certificate, a copy of your passport or ID-card and the state fee by Postal Order (cheque from a post office; payable to Estonian Embassy).
The loss of an Estonian driving license has to be reported to the Estonian Road Administration (Maanteeamet) firstname.lastname@example.org.