Update 11 January 2021 see therefore my doubt viz Coward Phaestos Ltd & Ors  EWHC 9 (Ch): Application, denied, for discovery /evidence order of data held in England, for use in Cypriot matrimonial proceedings. Held application could and should have been made in Cyprus under EU evidence regulation.
The ECJ has confirmed the subsidiary nature of the Evidence Regulation in ProRail, Case C-332/11. Regulation 1206/2001 does not govern exhaustively the taking of cross-border evidence, but simply aims to facilitate it, allowing use of other instruments having the same aim. The Court in this respect confirmed its view in Lippens.
Unfortunately the ECJ did not entertain the questions of the Belgian Hof van Cassatie on the role of the Brussels I Regulation, in particular Article 31 (confirming jurisdiction for provisional, including protective measures), Article 32 (defining ‘judgment’), and Article 33 (providing for automatic recognition and enforcement). Admittedly, the national court could have been more precise on the role, if any, it saw for these Articles of the Jurisdiction Regulation in the dispute at issue (the appointment of an expert, for an expertise to be carried out outside of Belgium).
A possible impact of Articles 31-33 JR might relate to challenges of the carrying out of work by the expert (e.g. by way of seeking an injunction) in The Netherlands: would the Dutch court be prevented from second-guessing the expert’s brief, qualifications etc?
The Hof van Cassatie itself, according to the order of reference to which JÄÄSKINEN AG refers, flagged the Jurisdiction Regulation at the instigation of ProRail, whose appeal alleged infringement of Regulation No 44/2001. ProRail sought to infer that the power to order that an expert’s report be obtained lies exclusively with the courts of the place in which it is to be executed and, a contrario, that such a measure does not have any extraterritorial effect except by virtue of the authorisation of the Member State in which that investigative measure is to be carried out. Switzerland, intervening on this point because of the similarities with the Lugano Convention, suggested that the measure by which a court charges an expert to carry out an investigation in the territory of another Member State is neither a provisional nor a preventive measure within the meaning of Article 31 of Regulation No 44/2001, on the ground that such a measure cannot produce extraterritorial effects, nor a judgment which may be the subject of recognition or enforcement within the meaning of Article 32 of the same regulation.
In the end, the AG suggested that the reference to the Jurisdiction Regulation was not specifically enough included in the questions put to the ECJ – a view which the ECJ implicitly agreed with. An interesting opportunity wasted, so it would seem.