To understand how the situation might change post Brexit, one first has to consider the status quo. Council Regulation (EC) No 1215/2012, aka Brussels Recast, aka Judgment Regulation Recast, aka Brussels II, determines matters of jurisdiction for EU members apart from Denmark who opted out of the Regulation. As an EU regulation, Brussels II has direct effect in the UK. However, Brussels II is not universal in that “if the defendant is not domiciled in a Member State [jurisdiction shall be] determined by the law of the Member State” (article 6.1). Accordingly, Brussels II only forms part of the patchwork of English Law on jurisdiction, along with the Brussels Convention of 1968, the Civil Jurisdiction and Judgements Act of 1982, and the common law.
To consider how English law on jurisdiction might change upon Brexit it is necessary to very briefly consider the history of the Brussels regime on Jurisdiction. The Brussels Convention of 1968 was given the force of law in England by section 2 of the Civil Jurisdiction and Judgements Act 1982. The Brussels Convention was superseded by Brussels I and subsequently Brussels II on 10 January 2015, but section 2 of the 1982 Act, which brings the Brussels Convention into English law, remains in force.
Therefore, assuming that the article 50 agreement dealt only with the terms of exit and not the terms of the new relationship between Britain and the EU, the Brussels II void – caused by the Regulation losing the force of law upon the repeal of the European Communities Act 1972 – would automatically be filled by the Brussels Convention via section 2 of the 1982 Act. If the Brussels Convention were to replace Brussels II, even if only as a temporary stop gap while negotiations as to the transitional provisions and substantive new agreement between the EU and Britain were taking place, much would stay the same; the general rule that a defendant is to be sued in their own domicile would remain, the rules relating to additional jurisdiction, such as the rule in tort claims that a claim can be brought where the harmful event occurred, would largely remain familiar. However, there would be some very noticeable changes; whereas a personal injury claimant (a person injured by the policyholder) can sue an insurer in his or her domicile under Brussels I and II (where the foreign law permits a direct action against the insurer) the Brussel Convention does not create this additional jurisdiction. In other words, even where the foreign law permitted direct actions against the insurer, the insurer could challenge jurisdiction where the accident occurred outside England and Wales and the insurer did not have a domicile in England and Wales.
It is possible that Britain and the other EU Member States might agree on an alternative arrangement in respect of the law of jurisdiction during the article 50 negotiations but I consider this unlikely. I consider it more likely that an alternative arrangement, if there is to be one, would form part of the agreement to set the terms of the new relationship between Britain and the EU which may not occur for several years after Brexit, via article 50, has been concluded. Possibilities for the long term include Britain becoming a Contracting State to the Lugano Convention of 2007 to which Denmark, Iceland, Switzerland and Norway are Member States. The Lugano Convention of 2007 is virtually identical to Brussels I but does not incorporate the changes made in Brussels II such as the changes to combat the Italian torpedo (issuing a claim in Italy to stymie the possibility of bringing a claim in another jurisdiction where the wheels of justice move faster).
In my opinion it is likely that some variant of the Brussels regime, first conceived in the Brussels Convention of 1968, will continue as government, business, individuals and insurers will all likely be keen to maximise legal certainty, which augurs against any radical change. Further still, it appears likely that both EU Member States and Britain will be keen on something like the status quo. However, as ever, the devil will be in the detail, and if, as seems likely, the article 50 agreement will simply cover the terms of the exit as opposed to the terms of the new relationship between Britain and EU, it seems possible, if not likely, that Brussels II will be replaced by the Brussels convention, at least in the short term, which has some significant differences to the Brussels II regime. The longer term is inevitably much more uncertain, but the experience of other countries in Europe but outside the EU would suggest that Britain may well become a signatory to the Lugano Convention. Indeed, until repeal on 1 January 2010, section 3A of the 1982 Act gave the Lugano Convention the force of law in England and Wales, so its reintroduction would hardly be a major change.
 See Article 288 of the Treaty on the Functioning of the EU, which specifies that EU regulations are directly applicable in EU countries (as confirmed by the Court of Justice of the European Union in the judgment of Politi v Ministero Delle Finanze Case C-43/71).