This will depend on national law, but you could in principle raise this issue in the Executing State with the Executing authority, or with the authority responsible for the substantive criminal case. The former might not be competent to decide on the conflict, but you can request that your application for consultation proceedings between the Member States be forwarded to the competent authority.
Simultaneously, or alternatively, the ISL (or the EU third state lawyer) may lodge a request with their competent national authorities for consultation proceedings to be launched.
You could also consider lodging a request to the relevant National Member at Eurojust. Eurojust is a judicial cooperation unit composed of national prosecutors, magistrates, or police officers of equivalent competence, detached from each Member State according to their own legal systems: see Council Decision 2002/187/JHA on setting up Eurojust, as amended by Council Decision 2003/659/JHA, and Council Decision 2009/426/JHA (16th December 2008) on the strengthening of Eurojust[24]. It is “particularly well suited to provide assistance in resolving conflicts of jurisdiction” (Recital 14 CJ Framework Decision). In practice it has extensive experience on the matter.[25] Although consultation proceedings are primarily between the national authorities involved, they may take place with the assistance of Eurojust and where no agreement can be reached, the case should in principle be referred to Eurojust (Recitals 4, 10 and 14 and Article 12 CJ Framework Decision).
Eurojust, through its National Members or the College, can contact national competent authorities and draw their attention to a possible conflict and establish consultation proceedings under its auspices (Articles 82(1)(b) and 95(1)(c) Treaty on the Functioning of the EU and Articles 6(1)(a)(ii) and (c) and 7(1)(a)(ii) and (c) Eurojust Decision). The College may in certain situations issue a non-binding opinion on which jurisdiction should prosecute (Article 7(2) Eurojust Decision). Eurojust may also assist in cases of multiple EAWs (Article 16(1) and (2) EAW FD).
National Members of Eurojust must be informed by national authorities of cases where conflicts of jurisdiction have arisen or are likely to arise (Article 13(7)(a) Eurojust Decision).
Despite the absence of any provision establishing a right of the concerned person to trigger Eurojust’s intervention, the fact that national authorities must inform National Members of possible conflicts and that Eurojust may intervene, should be a sufficient legal basis for you to address a request to Eurojust, and for the relevant National Member to take action.
You can also attempt to trigger Eurojust’s action indirectly by requesting your competent national authority (eg. the Executing State authority in an EAW case) to forward the case to Eurojust. The ISL may also trigger Eurojust’s action directly or indirectly via her competent national authorities.
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[24] For more information, see Eurojust’s website.
[25] See a report on Eurojust’s casework on conflicts of jurisdiction drafted for the Eurojust Strategic Seminar on Prevention of Conflicts of Jurisdiciton.