Emerald is the operator of the Carrickfinn PSO and has to have an EU AOC to bid and fly that - they couldn't do that on a UK AOC and if they have designs on other operations like DUB-Rennes which Aer Arann/Stobart did for several years, they would need the EU AOC there also.
However you can have a UK base without having a UK AOC. Ryanair has several UK bases at which it has EI- aircraft only and SAS Link is an Irish-registered operator flying on behalf of SAS from a Heathrow base. You just can't fly UK domestics or from the UK to a non-EU country (unless air services agreements permit) on that basis.