The UK employer is obliged to register as an employer here in Ireland and obliged to deduct Irish PAYE etc from your wife's gross salary.
They will have to have an official registered address which is their "branch" address in this State - you could use address as a last resort but ideally they should get a payroll company to do the payroll, they would probably do it for a nominal fee. Then they operate Irish taxes on her salary as normal and pay the PAYE/PRSI/USC over to Revenue on a monthly basis. They also have to submit a P35 to the Irish Revenue every year with your' wife's salary details on it.
Here is the Revenue link stating that employments performed inside the State are liable to Irish PAYE
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I have done this for several UK companies who employ sales reps in Ireland