Update II 8 April 2020 : just after my earlier update, Lecta Paper’s scheme of arrangement sanction hearing was published on BAILII: Trower J considers the end of the transition period at 40.
Update 8 April 2020 compare Crossley & Ors v Volkswagen Aktiengesellschaft & Ors  EWFC 28 in which Waksman J succinctly holds at 12
‘On 31 January 2020, shortly after the conclusion of the trial, the UK withdrew from the EU. I asked the parties whether this would make any difference to any of the arguments made before me. I received written submissions on that topic on 4 and 9 March 2020. It is clear from those submissions that it is common ground that Brexit makes no difference here because EU Law (including the jurisdiction of the CJEU) will continue to have effect as if the UK was still a Member State until the end of the “transition period” which is 31 December 2020. This includes the general obligations which the UK owes qua Member State. Accordingly, for the purpose of this judgment, the position is as it was before 31 January 2020.’
(update ctd) and note VB v TR  EWFC 28 where Mostyn J at 6 refers to the (until the end of the transition period) exclusive EU external competence to recognise (on the basis of reciprocity) countries as being on the accession list for the Hague 1980 Hague Convention on the Civil Aspects of International Child Abduction – which Bermuda is not even if the UK itself had recognised it as being part of the Hague regime. At 12: ‘In my judgment the law needs to be changed as between the United Kingdom and its overseas territories to provide that the 1980 Hague Convention operates between them. It is an embarrassment that if a child were taken from Bermuda to United States of America the Convention would apply, but if the child is brought here it does not. Alternatively, the law needs to be changed so that the automatic recognition of orders within the United Kingdom under the Family Law Act 1986 is extended to orders made in the Overseas Territories and the Crown Dependencies.’
In Bournemouth, Christchurch and Poole Council v KC et al  EWFC 20, Dancey J at 62 ff is the first UK judge to my knowledge to discuss the implications of the UK’s separation from the EU’s civil procedure /justice and home affairs agenda, particularly in the transition period. It includes a discussion of the UK’s Brexit (EU Exit) Regulations 2019/2003, reg 3, and the European Commission notice on transition provisions.
The care proceedings concern W, a girl aged 9, nearly 10. W’s parents, who were married, are Polish nationals and W was born there. Following the separation of the parents in Poland in April 2016, contested contact proceedings there resulted in an order providing that W live with the mother with contact to the father. The father’s parental responsibility was limited to decisions about medical treatment and education. Following the breakdown of the father’s contact with W, the mother brought her to the UK in June 2018 where they have remained since. That was done without the father’s agreement, although he was aware the mother planned to relocate and acquiesced once the move had taken place. The mother did not tell the father of her and W’s location within the UK.
The legal framework, therefore, is Brussels IIa, Regulation 2201/2003. Dancey J at 63 concedes that by reg 8 of 2019/2003, dealing with saving/transitional provisions, the UK’s revocation from Brussels IIa does not apply to proceedings before a court in a Member State seised before 31 December 2020. However he then refers to the EC Notice to Stakeholders: Withdrawal of the United Kingdom and EU Rules in the Field of Civil Justice and Private International Law: 18/1/2019, and suggests it means that EU rules on recognition and enforcement will not apply to a UK judgment, even if the judgment was given, or enforcement proceedings started, before 1 January 2021 unless the judgment has been exequatured (declared enforceable by the courts of the Member State where recognition or enforcement is required) before 1 January 2021. Support for his opinion is found I suspect mostly in Heading 2.2 of that Notice – although I should warn against the matter of factly nature of that notice without much justification given for the positions it takes. Update 29 03 2020 and as Xavier Lewis has pointed out to me, the notice was put out there in the eventuality of a no deal – Dancey J does not refer to the Withdrawal Agreement at all.
At 66 Dancey J suggests in practice the consequence should not be too dramatic in the case at issue for ‘one or other of the parents should apply promptly in Poland for a declaration recognising this judgment and the order that will follow (exequaturing the judgment).’ That absence of real delay in the case at issue may well be true (it is confirmed by a letter from the Polish consulate) however the implications are already clear and no surprise. Enforcement of UK judgments will be a lot less smooth post Brexit.