I'm a bit confused that you say you hold an EASA licence, but then talk about the IAA not accepting a JAR-FCL licence.
If you hold a licence with Part-FCL written on it, it is an EASA licence that the IAA or another State will replace with one of their own.
If you have a licence with JAR-FCL written on it the IAA won't convert it because of what the European Aircrew Regulation says:
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Article 4
Existing national pilots’ licences1. JAR-compliant licences issued or recognised by a Member State before 8 April 2012 shall be deemed to have been issued in accordance with this Regulation. Member States shall replace these licences with licences complying with the format laid down in Part-ARA by 8 April 2017 at the latest.
------------------------------So legally a JAR-FCL licence became a Part-FCL licence on 8th April 2012. As the CAA is funded entirely by the charges it makes to applicants for what it does, it applied a charge for replacing a JAR-FCL licence with a Part-FCL licence. This would have to happen on application by pilots or whenever any change was made to the licence (e.g. adding a rating or change of address) because the CAA couldn't issue or re-issue a JAR licence after April 2012, so I guess they expected all the licences would have been replaced within the 3 year period. I guess now a pilot with a JAR-FCL licence could argue that the CAA is legally required to give them a Part-FCL licence immediately in exchange for the JAR licence, because the EU Regulation says - "Member States shall replace these licences with licences complying with the format laid down in Part-ARA by 8 April 2017 at the latest".
Theresa May's deal includes being bound by the rules of the EU and it's institutions and agencies (including EASA) for the Brexit transition period. So if her deal gets through Parliament the CAA can carry on issuing Part-FCL licences for that period. If it's a "no deal" Brexit the UK drops out of EASA immediately. In that case all licences ever issued by the CAA, including Part-FCL licences, will become UK national licences that will not be valid to fly aircraft that are registered in other European countries. To continue to do that a UK licence holder will need a validation of the UK licence from another Member State - and then a licence conversion. For the conversion a CPL or ATPL holder will have to take all their exams again with another Member State.
The European Commission has published an open letter stating very clearly that all Part-FCL pilot licences issued by the UK CAA will lose their European privileges on Brexit day if it's no deal.