Motorists resident in Ireland with a UK, including NI, driving licence are advised to exchange that licence for an Irish driving licence before the UK withdrawal date from the EU. In the event of a no-deal Brexit, the UK will no longer be a Member State and so the UK (including NI) driving licence will not be recognised. People resident in Ireland will no longer be able to drive on a UK (including NI) driving licence.
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.
In the Road Traffic Acts, legislation exists to allow for the recognition of foreign driving licences for exchange purposes. In a no deal Brexit, when the UK becomes a third country, arrangements can be made under those Acts. Ireland will pursue this option which will allow UK (including NI) licences to be exchanged for an Irish licence. However this may take a little time to complete as it involves a formal agreement and legislation here in Ireland.
As the Deputy is aware, historically Ireland has not issued driving licences to asylum-seekers.