An Irish company can certainly carry out a contract in the UK. There is no need to register a UK company.
If the people carrying out the work are existing employees of the Irish company on secondment to the UK with PAYE etc paid through the Irish payroll then on a short term basis you can continue with this arrangement. It would be different if you take on workers in the UK then your Irish company would have to register as an employer in the UK.
VAT is a whole other ball of wax.
Perhaps most importantly you also have to worry about UK Relevant Contracts Tax. Your client will most likely deduct RCT from the payments, it will be up to you to reclaim this. I suggest a call to UK Revenue.