All EU civil judicial cooperation regulations will continue to operate vis-à-vis the UK during the Brexit Transition Period (to 31 December 2020, unless extended). 

Here is a brief summary:

• Allocation of jurisdiction as between the UK and other Member States continues to be governed by the recast Brussels Regulation 2012, which also governs the enforcement of UK judgments in EU states and of incoming EU judgments. 
• The UK is also still signed up to the Hague Choice of Court Convention, as an EU Member State, because, although not a member of the EU, it is treated as though it were under the Withdrawal Agreement. But where the relevant parties are in the UK and another EU state the Brussels Regulation will take precedence over Hague.
• Both the Rome I and Rome II Regulations on Applicable Law (for contract and tort respectively) continue to apply.
• Service of all court documents out of the jurisdiction into other EU states is governed by the Brussels Service Regulation (under which service by post from one EU state to another is only permitted through an "authorised agency" (for the UK this is done via the Foreign Process Section of the RCJ)); but service by a contractually agreed method within the jurisdiction is permitted under the Civil Procedure Rules (so as to avoid having to serve claims out of the jurisdiction).

The Transition Period may be extended under the Withdrawal Agreement (WA) but the government has legislated not to permit the UK government to agree an extension with the EU. So UK primary legislation will be required to permit an extension. The WA provides that any extension must be agreed before 1 July 2020 and can be for up to "one or two years".

We don't know for sure what any agreement as to the UK's future relationship with the EU after the Transition Period as regards civil judicial cooperation will look like, but, on 28 January 2020, Lugano Convention States, Norway, Iceland and Switzerland indicated that they support the UK acceding to Lugano, in its own right post Transition. The agreement of the EU and Denmark would also be necessary for the UK to accede. The Lugano Convention allocates jurisdiction, and deals with the enforcement of judgments between EU and EFTA Member States, and Switzerland, in a similar way to the Brussels Regulation, but without the refinements that were introduced to the Regulation in 2012. The UK remains a party to the Lugano Convention during the Transition Period because it is still treated as though it were an EU Member State.

There is still a possibility of No Deal with the EU on these issues at the end of the Transition Period. There are statutory instruments (passed in 2018 and 2019) which will cater for gaps if there is No Deal at that point.

Decisions of the CJEU remain binding in the UK until the end of the Transition Period. Any case pending before the CJEU at the end of the Transition Period, involving the UK, will also be binding. Thereafter, the Supreme Court may depart from EU law (as it stands at the end of the Transition Period).  In consultation with the senior judiciary, a government minister may designate other courts that may do so (a provision that received significant criticism during its passage through Parliament).