Report based on country
Welcome to CivICRM SE! You can get a report for the Republic of Ireland using the standard reports and filtering on Country, but CiviCRM doesn't have a field for the countries withing the UK by default. I don't think it would make sense to edit the Country field in the postal address unless you will never have any postal addresses outside the UK/ROI.

You will have to create a custom field for this. Obviously you can enter this manually when you set up the contact. You could populate this automatically from the postcode, but this would require some lookup from postcode to UK country (for example https://www.robertsharp.co.uk/2017/08/09/a-table-that-shows-the-uk-region-for-all-postcode-districts/ has the mapping) so would require some coding. There is a postcode lookup extension https://civicrm.org/extensions/uk-postcode-lookup but I think that's for looking up addresses and won't provide the UK country. Also be aware that postcode isn't the same as administrative area. For example Hay-on-Wye is in Wales, but has an English postcode. Hope that helps.