UK Inheritance Tax is not normally paid by the recipient of a gift or legacy so you won't have any IHT to pay in the UK. It can only be paid by the recipient if the deceased estate does not have sufficient funds to pay it on a gift made in the seven years before the death of the donor.
If you are UK tax resident, you may have CGT to pay in the UK but any CGT paid in Eire will be deductible from your UK liability on the same gain. The first £11,100 of gains made by an individual are tax free. Take a look here for more information on CGT. If you aren't a UK national, you can choose to pay tax on the remittance basis if you wish which means that you would only pay UK tax on UK source income and gains and on non-UK income remitted to the UK. Take a look at section 9 of RDR1 here.
I hope this helps but let me know if you have any further questions.