The UK's Department of Transport says Irish and UK licenses will be recognized on either side of the border if a no-deal Brexit happens. However, Northern Ireland license holders may need a permit to drive in other EU countries.

The Department said UK or Northern Ireland driving licence holders will not require an International Driving Permit (IDP) to drive in the Republic of Ireland post-Brexit. 

However, in the event of the UK leaving without a deal, UK and NI driving licence holders may need an International Driving Permit in addition to their driving licence to drive in other EU or EEA countries.

The Road Safety Authority has also reminded British citizens living in the Republic that their license needs to be exchanged for an Irish one within 100 days of coming to live here. This has been the case for some time and is not as a result of Brexit.

The authority has issued a question and answer briefing for the most commonly asked questions about Brexit and licenses: 

Q. I live in the UK and hold a UK driving licence and would like to be able to drive a vehicle in Ireland from time to time on holidays after Brexit.

A. As a visitor there is no reason that you will not be able to drive in Ireland for holidays with your existing driving licence. This will apply regardless of whether there is a deal or no-deal Brexit. You should not be required to carry an International Driving Permit with you in order to drive here, just ensure to carry your UK driving licence with you.

Q. I live in Ireland, hold a UK driving licence and would like to be able to continue to drive here in Ireland after Brexit on my UK driving licence.

A. In a no-deal Brexit scenario, as a resident in Ireland, your UK driving licence will not be valid to drive here in Ireland. Prior to 29 March 2019 you could decide to exchange your UK driving licence for an Irish driving licence- the usual processing timeline for this is quite short- up to ten days. However, that may change in the event of a high level of demand for driving licence exchanges. More information on exchanging a driving licence can be found here. It is also the case that, following a no-deal Brexit, as things stand your driving licence will not be valid for exchange following Brexit. There is a possibility in such a scenario that Ireland and the UK will enter into a bi-lateral arrangement on driving licences where Ireland recognise UK driving licences. Inevitably, this will take a little time to complete as it involves a formal agreement and legislation here in Ireland. The position is different if there is an agreed Brexit. The current Withdrawal Agreement provides for arrangements to recognise UK driving licences across the EU during the transition period which is until December 2020. It is expected that negotiations on the future relationship, which will take place during the transition period will seek to put a permanent arrangement in place. This means in an agreed Brexit, for residents in Ireland, that UK driving licences in Ireland will be recognised until December 2020 and subject to whatever agreement is then reached in the transition period.

Q. I'm a UK citizen, I live here in Ireland and I hold an Irish driving licence can I drive in the UK from time to time?

A. An Irish driving licence will continue to be recognised in the UK. An Irish driving licence is an EU licence and the UK has indicated it will continue to recognise EU driving licences.

Q. I live in Ireland and drive trucks across Europe including the UK as part of my job, I am the holder of an Irish driving licence with truck categories, can I drive on the Irish driving licence in UK after Brexit?

A. An Irish driving licence is an EU driving licence and the UK has indicated it will continue to recognise EU driving licences.

Q. As a UK licence holder driver where can I find further information to prepare to drive in the EU after Brexit?

A. https://www.gov.uk/government/publications/prepare-to-drive-in-the-eu-after-brexit