If the UK leaves the EU on 29 March 2019 and there is no Withdrawal Agreement, the EU laws will cease to apply at 11pm and the UK will be relying upon existing national law and international laws.
The existing divorce jurisdiction which is currently governed by the EU law will simply continue without there being any need for a change in the procedure as it is similarly reflected in our national law.
Approximately half of the EU member states are signatories to the 1970 Hague Convention. The Hague Convention allows UK court orders to be enforceable in other member states. The member states that are signatories to the Hague Convention are likely to continue to recognise a divorce in England and Wales.
However, there are 15 states who did not sign the Convention or who have signed but not ratified it. It's hard to imagine why these countries would not recognise divorces when they have done previously, but there are no guarantees moving forwards.
If you are a national or have assets such as property in one of those 15 countries, you may have to bring fresh proceedings not only in relation to the divorce but also associated financial matters, in multiple countries. For example, if you are French, your spouse is Spanish, you have assets in both of those countries but live together in the UK, you may have to issue divorce proceedings in all three countries, in order for your divorce to be recognised and for you to be able to obtain an enforceable order regarding those foreign assets.
If you are concerned that your divorce may not be recognised in those member states, it would be worth trying to obtain the decree absolute before the UK leave the EU.
A decree nisi is a provisional decree of divorce granted by the court if it is satisfied that you have met the requirements to obtain a divorce. This means your marriage still exists and you are not yet divorced. A decree absolute is the legal document that ends your marriage. You need to wait at least 43 days after the date of the decree nisi before you can apply for a decree absolute.