The United Kingdom driving licence of a person who settled in Estonia before the end of the Brexit transition period (31 December 2020) will remain valid for 12 months after the end of the transition period, i.e. until 31 December 2021.
Such a driving licence will be exchanged on the same terms and conditions that apply to driving licences of other EU Member States only during the transition period, i.e. until the end of 2020. You can view the detailed terms and conditions here. In 2021, the exchange will be subject to the same rules that apply to driving licences of persons from other countries of the 1968 Vienna Convention on Road Traffic. This is described in more detail here.
The United Kingdom driving licence of a person who settled in Estonia from 1 January 2021 will remain valid in Estonia for 12 months from the date of settling in Estonia. If after that period the person wishes to drive a vehicle in Estonia, they will have to exchange their United Kingdom driving licence for an Estonian driving licence. You can view the detailed terms and conditions of such exchange here.
Reference: section 2641 of the Traffic Act.