Immediately after Brexit had taken place (for my assessment of what is going to change see here in English and here in German, the European Commission set up a new site: The European Union and the United Kingdom – forging a new partnership. Today, it published draft negotiating directives, as a basis for its negotiation mandate.
The draft directives are based on the Political Declaration that was agreed between the European Union and the United Kingdom alongside the Withdrawal Agreement. They cover trade and economic cooperation, law enforcement and judicial cooperation in criminal matters, foreign policy, security and defence, participation in EU programmes and further areas of cooperation.
I ran a quick search and it appears that judicial cooperation in civil and commercial matters does not feature at all in the negotiation directives. This may be a good thing, if one assumes that the reason for not including it is that the Commission does not believe it is going to controversial and an agreement with the United Kingdom on an adequate replacement of the Brussels Regulation regime would be a no-brainer. On the other hand, it would be quite concerning if judicial cooperation in civil and commercial matters were a matter that simply is so low down the list of priorities that it will not be included in the core negotiations at all.
The term “judicial cooperation” only appears in the context of criminal matters. The European Commission proposes (as it has done before) to tie the law enforcement cooperation and judicial cooperation in criminal matters to the United Kingdom’s continued membership in the European Convention of Human Rights:
“The envisaged partnership should be underpinned by commitments to respect fundamental rights including adequate protection of personal data, which is an enabler for the cooperation. In this context, the envisaged partnership should provide for automatic termination of the law enforcement cooperation and judicial cooperation in criminal matters if the United Kingdom were to denounce the European Convention of Human Rights (ECHR). It should also provide for automatic suspension if the United Kingdom were to abrogate domestic law giving effect to the ECHR, thus making it impossible for individuals to invoke the rights under the ECHR before the United Kingdom’s courts.” (#113 of the draft negotiations directives).