I admit that I'm fumbling in the dark here but I hope that I might at least be able to point you in the right direction as to where to get reliable information from.
As I read this Bar Council document, the transcripts will be treated as if they're copyright sound recordings:
https://www.barcouncil.org.uk/media/499359/copyright_status_of_court_judgments.pdf
(That document specifically deals only with judgments, rather than full proceedings but the principle would appear to be the same).
Further, it would seems that
(a) the copyright holder is the Crown ; and
(b) the Controller of Her Majesty’s Stationery Office is responsible for administering that copyright.
The National Archives website indicates that the majority of Crown copyright material may be reproduced under the terms of the Open Government Licence:
http://www.nationalarchives.gov.uk/doc/open-government-licence/version/3/
So the question that needs to be asked is to whether court transcripts are covered by that licence. The place to ask that question would appear to be here:
https://www.nationalarchives.gov.uk/contact-us/public-sector-information-enquiry/
I hope that makes some sort of sense and points you in the right direction.