As far as I know, it was the Wreck Commission. I don't know if they kept specific records of who wrote what. It was probably written down by some very overworked secretaries, but that's just a guess on my part.
The report itself was written by Lord Mersey, no doubt with input from his assessors.
The evidence recorded in the transcript is the work of a private firm that specialised in stenography.
For the record, the US report was written by Senator Smith, who accepted a few suggested small changes from members of the committee. The transcript is mainly the work of Bill McKinstry, an experienced stenographer.
It's interesting that no mention of McKinstry, or any other stenographer (presumably more than one worked on the testimony?), is included in the Senate Document of the transcripts. However,there is mention of two secretaries - Charles G. Bennett and H.M. Rose. I have no idea what their connection with the hearings is.
I recall reading somewhere that Mersey's son had a hand in the creation of the British Report. Is there any veracity to this rumor?
Mersey's son, Clive Bigham was appointed the secretary to the court, but that merely means that he organised it. As in the USA, 'secretary' doesn't always mean a pen pusher. Clive Bigham was a brilliant multi-lingual scholar and his appointment was not just nepotism at work.
McKinstry did most of the work, but he could not have done it all, as on one evening several senators took evidence separately. The interview with Fred Barrett is in a different layout from the rest, so maybe that was also the work of some other stenographer. I wonder if the men you mention were responsible for the general paperwork of the committee.