VAT recovery depends solely on whether or not the supplies of goods or services are then put to a taxable use
Here is a quote from section 12 VAT Act 1972:
"In computing the amount of tax payable by him in respect of a taxable period, a taxable person may, insofar as the goods and services are used by him for the purposes of his taxable supplies or of any of the qualifying activities, deduct [, subject to making any adjustments required in accordance with section 12D]1-"
If you are not VAT registered the UK supplier may have an obligation to charge VAT
If you can recover the VAT then its recovered from Irish revenue not the UK revenue.