I think it helps to start with our current position. At the moment, the EU has a set of rules that determines (a) who has jurisdiction (the authority to hear a case) and (b) the ability to enforce judgements in a member state. That’s called the Brussels Regulation.
It’s very extensive; the idea behind it is to stop ‘parallel proceedings’ (the same case being brought in different member states) and ‘irreconcilable judgements’ (conflicting judgements in different courts).
There’s been a fair bit of tension between the UK and EU regulation, but I won’t go into that here. The takeaway is that if an English court were to make a judgement, a party wouldn’t be able to pursue the case in another member state; the foreign court would respect the decision of the English courts.
Now that we’re leaving the EU, we’re no longer going to be governed by the Brussels Regulation. And as the question rightly says, we’d want to negotiate agreements so other member states respect decisions made by the English courts. There’s currently two existing conventions which we could join – the Hague and the Lugano convention, or we could negotiate our own agreement (like Denmark). If we did, I think it’s likely we’d have similar provisions to what we have now.
To answer your question, yes – if we didn’t have an agreement with the EU it would be a problem. If there were no rules, a French court wouldn’t have to respect judgements made by the English courts, nor would it be required to prevent parallel proceedings. In my view, that all seems very unlikely; the EU cares a lot about certainty and harmonisation, and in theory, we wouldn’t have to respect their judgements either – so no-one would benefit there.
Finally, with respect to the last point, I think it’s important to remember that the London Commercial Court has a lot of respect for reasons not to do with the EU. Foreign parties choose the court because it’s seen as fair, efficient and commercially aware. The court respects contracts (which companies like) and they’ve got great weapons at their disposal i.e. the power to freeze foreign assets and to stop parties bringing cases in other courts.
I don’t know what kind of agreement we’ll end up negotiating, but I doubt it’ll be too dissimilar to what we have now.The EU has a lot of leverage, but they’d also respect the power of the courts we have.