UK & Europe
Although construction disputes are affected by Brexit, any disruption will be temporary, since the UK can fall back on new global rules on jurisdiction and enforcement in the longer term. Even now, some aspects of dispute resolution (including governing law and arbitration) are hardly affected by Brexit at all.
Much has been written about the effect of Brexit on commercial disputes, including those in the construction industry. However, its main impact is on disputes arising from older contracts, not ones agreed after the Brexit transition period, which came to an end on 31 December 2020. Even where there is a 'Brexit effect', problems can often be remedied by obtaining an English anti-suit injunction (suppressing foreign proceedings that breach an exclusive English jurisdiction agreement) or by enforcing judgments under bilateral arrangements or local procedural laws. A practical example illustrates the position well.
Suppose an English company is contracted to construct a block of flats in London for an employer based in Italy. The contractor makes slow progress, and the employer terminates the contract. The contractor reacts by suing the employer in its home courts (in Italy), even though the contract gives exclusive jurisdiction to the English courts. The employer, on the other hand, seeks compensation from the contractor's parent company in Germany under a separate guarantee, issuing separate proceedings in the English Technology & Construction Court ('TCC'). Like the main contract, that specifies English courts, giving them exclusive jurisdiction. Both documents are said to be governed by English law.
How does Brexit affect these proceedings and the enforcement of any judgments that might result from them? The answer depends largely on timings.
If all the events described took place before the end of the Brexit transition period, all jurisdiction issues would be would be determined by the Brussels Regulation (Recast) EU 1215/2012 ('the Recast Regulation'), which applied to the UK up to 31 December 2020 as if it were still an EU Member State. Under the Recast Regulation, English courts could try both claims straightaway, even though the Italian proceedings were initiated first. This is because the choice of any courts in the EU is supported by Art 25 of the Recast Regulation, and where a jurisdiction agreement is exclusive, this takes precedence over rules designed to avoid parallel proceedings (Art 31(2)). It therefore makes no difference in this example that the Italian courts were 'seised' first: they would be obliged to stay proceedings.
The choice of English law would also be supported by governing law rules set out in the Rome I Regulation EC 593/2008 (Art 3).
If the dispute arose now
The situation would be rather different, however, if a dispute of this kind were to arise now. The main contract and guarantee would have been concluded before the end of the Brexit transition period, but the UK is now a 'third country' from the EU's point of view, and so does not benefit from the jurisdiction and enforcement rules contained in the Recast Regulation. As a result, the Italian courts might well not recognise the exclusive jurisdiction agreement in the main contract. On the contrary, they would be obliged to try the case on the basis that they are in the domicile of the defendant in the case - the Italian employer (Art 4). This is convenient from the contractor's point of view, because any resulting judgment is easy to enforce against the employer, despite Brexit, since this can be done within Italy and no cross-border issues arise.
However, from the contractor's point of view, there are two flies in the ointment. One is that this will not stop the action under the guarantee going ahead in the TCC, although under common law rather than European jurisdiction rules. The second is that it should prove relatively easy for the Italian employer to supress the Italian proceedings by obtaining an English anti-suit injunction. Since the contractor is an English company, it would not be able to ignore the injunction, even if the Italian courts were willing to proceed. (This is an example of how English courts are expected to flex their jurisdictional muscles now that the UK is fully out of the EU, and how this will have a major impact on some cross-border disputes.)
Meanwhile, the choice of English governing law would be respected by both the Italian and English courts, regardless of Brexit. This is because the Rome I Regulation, which is still followed by courts in the EU, of course, does not discriminate between the choice of systems of law that originate within the EU and those that originate elsewhere (Art 2). The Rome I Regulation has also been copied into domestic UK law, with minimal changes, so it will continue to be applied by UK courts too.
The upshot is that, although the situation is messy, English courts still end up trying both the main dispute and the claim under the guarantee, and applying English law as they go. However, problems may well arise further down the line, if and when the employer is successful and needs to enforce an English judgment against the parent company (guarantor) in Germany. Before Brexit this could be done quite easily, under Chapter III of the Recast Regulation, but now it is possible only under old bilateral enforcement arrangements or the German civil procedural code. In both cases the procedure is relatively complicated and there are several opportunities for the German parent-guarantor to be obstructive, so local advice would have to be taken as early as possible to identify the best way forward.
If the dispute arose in ten years' time
Although Brexit will have a disruptive effect in the short term, as we have seen (and this may continue for some years) the effect will diminish over time. This can be seen if we consider the same scenario arising in ten years' time. At that point, the legal position would be simple again. Instead of European rules obliging the Italian court to decline jurisdiction over the initial dispute, the Hague Convention on Choice of Court Agreements 2005 ('the Hague Convention') would do this - provided the construction contract and guarantee were entered into no earlier than 1 January 2021, which is when the UK re-joined the Hague Convention after Brexit. It should also be possible to enforce an English judgment on the guarantee quite straightforwardly in Germany (Hague Convention Art 6 and Chapter III).
This is, of course, no comfort to parties with disputes based on older contracts, or no contract at all, but it does mean that anyone entering into a new contract can rely on the Hague Convention, provided it gives the chosen courts exclusive jurisdiction and does not concern excepted matters, some of which may be relevant to construction projects in the broader context (e.g. tenancies and certain aspects of insurance - see Arts 1 and 2 and the UK's and EU's Declarations).
Mind the gap!
The 'Brexit effect' on cross-border litigation is therefore time-limited. However, it is a gap in legal arrangements that any claimant needs to bear in mind if they are suing on an international contract dated earlier than 1 January 2021. Needless to say, most claims will fall into this time bracket for some years to come - so the gap is a wide one, and easy to fall into.
That said, none of these issues concerns jurisdiction or enforcement issues as between the UK and countries outside the EU/EFTA, and they also have no direct effect on arbitration, which is one reason why many parties have been including arbitration clause in their construction contracts in the lead up to Brexit. However, as mentioned already, the need to choose arbitration for this reason has now fallen away, given that the UK is now party to the Hague Convention again. So parties can go back to choosing English courts in most cases, if that is what they prefer. In other words, there may be many good reasons for choosing arbitration (privacy, excluding the right to appeal, and so on), but the 'Brexit effect' is no longer one of them.
 Also known in England as 'the Judgments Regulation'.
 Note the timing here: proceedings commenced just before the end of the Brexit transition agreement continue to be covered by EU jurisdiction rules, and judgments that arise from them will be enforceable under EU enforcement rules, regardless of when the judgment is actually given and enforcement proceedings begin, subject to normal limitation rules (Brexit Withdrawal Agreement Art 67). So the problems described here arise only because the 31 December 2020 (11pm GMT) deadline has been missed.
 The Rome I Regulation does not, however, apply in Denmark.
 Statutory Instrument 2019 No. 834
 In fact the UK was covered by the Hague Convention when it initially came into force in October 2015. However, that was in its capacity as an EU Member State. On Brexit, the UK was obliged to re-join as an independent contracting state, which it was able to do without the consent of the EU. The EU Commission's position is that, due to this change of status, the Hague Convention only applies to the UK now where the relevant jurisdiction agreement was concluded on or after 1 January 2021, or the relevant judgment flowed from such an agreement.
 It is true that the UK may be readmitted to the European fold where jurisdiction and enforcement matters are concerned. This would be achieved the UK re-joining the Lugano Convention 2007, which currently extends EU rules to three of the four EFTA states (Iceland, Norway and Switzerland). However, the EU may not allow this, and although the 2007 Convention is similar to the Recast Regulation in many respects, it does not allow the chosen courts to try a dispute straight away, even if they have been given exclusive jurisdiction, where proceedings have already begun in the courts of another contracting state.