Here are a few things (e.g. news reports) that document what platforms heights were and how that evolved. This is not in the numerous Railway Regulation Acts - those statutes are about who regulates and how, not the content of the regulations.
There was a somewhat farcical case in the Court of Session in 1869, brought by William Stewart against the Caledonian Railway Company. He had sprained his ankle, suffering permanent damage, climbing down from a train. He maintained that the platform was too low, so this was discussed in the reports of the case. The platform was said to be at "the common height of station platforms in Scotland" at 12-15 inches. However, that height was referred to as "low", though still suitable according to some experts. From the platform, there was a footboard 6 inches, a step 1 ft 10 in , and the floor 3 ft 2 in higher.
Discussions about the Regulation of railways Bill in 1886 included a proposal that the Board of Trade could issue orders that platforms should be raised to a specified height. However, this never made it into the act in 1889. Included in the press reports was data from the Caledonian Railway saying that their standard platform height was 2 ft 6 in, with a third of them at that level or above, up to 3 ft. So the trend was evidently upwards.
The Railway Archives has a version of the Board of Trade regulations for new railways showing the changes made in 1902. From then, platform height was to be 3 ft, or in very rare cases less but not below 2 ft 6 in. Previously the height had been 2 ft 6 in, but only as "desirable".
There are several other reports of raising platforms to the standard height around 1900, including a proposal at Cambridge in 1908, but with no heights given. However, it seems likely that this is when 3 feet became the norm, having been made a requirement for new build. Companies committed to making the change when rebuilding, and resisted demands (even from the MP for Haddenham) to do so otherwise. But the process was slow, with many small stations never needing their platforms rebuilt.