Many international businesses chose English law for their investments and deals. Except in certain limited respects, Brexit will not affect the substantive law of England. Accordingly, the terms of contracts governed by English law will not be interpreted differently post Brexit. There is therefore no reason to think that - market uncertainty aside - Brexit will cause commercial disputes to occur.
Many of the international rules regulating commercial disputes will also remain unaffected. The EU Rome I Regulation which regulates choice of law applies to EU and non-EU states alike. There is therefore no reason why Brexit would cause a court in another EU state not to respect the choice of English law in a contract.
International arbitration is outside the EU rules and governed by the New York Convention 1958, which most countries worldwide have ratified. LCIA and other international arbitration seated in England is therefore not expected to be affected.
There will be one significant change, post Brexit, England will be outside the Brussels regime which enables the service of legal proceedings, stays of parallel proceedings and enforcement of court judgments within the EU.
At the end of March 2017, the House of Commons Justice Committee recommended that, following Brexit, the Government should aim to replicate the Brussels regime as closely as possible, perhaps using the EU-Denmark agreement as a blueprint. Another option was to secure membership of the Lugano Convention, which extends a regime similar to Brussels to the EFTA states (Switzerland, Norway and Iceland).
This will be the subject of negotiations during the Brexit process. However, even if an agreement modelled on Brussels or Lugano is not achieved, we do not expect EU states to refuse to allow service of English court proceedings and enforcement of English court judgments, or vice versa. This is generally possible without specific international agreements. For example, the UK has no bilateral jurisdiction treaty with its largest trading partner, the USA, and this does not cause any major issues.