The HSE do issue public reports on major incidents. I am not sure if this will qualify for a multi-page PDF like some investigations, but we will know the outcome in either case.
The way they have been providing updates on this issue would be unusual for the vast majority of their work, so they have acknowledged public interest and the information releases will continue as there are developments. If there is a prosecution and court case, then we will also learn what has gone on via the usual media reporting of that.
What's important to realise is that these things take months or even years.
Yes they do, but normally only after criminal proceedings have concluded. One exception to this was the Buncefield explosion, but there were learnings from that incident and new knowledge about gas vapour explosions that was important to disseminate to the chemicals industry. But that is very rare (in fact unique in the time I have been involved in technical safety over the last 20 years) - I highly doubt they will issue any report on this incident before criminal proceedings have concluded (because a defendant can argue that it is prejudicial to a fair hearing and is against their human rights to a fair trial - not many people know that the human rights act also applies to companies).
It is not unusual for the HSE to issue public statements about notices being issued - after 28 days (the appeal period) those notices would be publicly available anyway (and they are published on the HSE website after about 7 weeks). But I bet you won't hear a dicky bird from them about what actually caused the crash until after any prosecution. We might get an inkling what could have occurred from any subsequent improvement notices that get issued. A Prohibition Notice is normally issued straight away, before all facts are known. Then subsequent Improvement Notices often get issued, and the content of those notices are normally a bit more informative - for example, they are normally accompanied by a schedule that details specific steps that need to be taken to improve health and safety, and you can often read between the lines what might have happened in the incident.
There are often public enquiries after major incidents - such as Piper Alpha, Kings Cross etc. - but they tend to be after mutiple fatalities have occurred. I highly doubt this will qualify.
Then there are also fatal accident enquiries - but as nobody was killed (thanfully) that won't be applicable to this incident.
So I would be highly surprised if there is any 'public' report issued on this incident by the HSE. However, it WILL be declared a major incident by HSE definitions because more than one person suffered a serious injury in the same accident.
There are some sectors of industry where an investigation report into a major incident has to be made publicly available by EU law (for example a summary report for all major incidents offshore will be made publicly available after legal proceedings have concluded from next month). But I believe theme parks aren't one of those industries.
The full facts will come out at any prosecution though, even if there is a guilty plea by a defendant. In such a situation, a 'narrative' is read out by the prosecution describing what happened, so that a judge can form a view on the seriousness of the offence for sentencing. The defence will then read out a mitigation statement, which attempts to get any potential sentence reduced somewhat (although I did hear of one case where the defence mitigation actually made things worse for them, and the actual sentence ended up being larger than expected!). But that narrative should be more than enough for the general public to be informed what occurred, which is why I don't think anything more will be issued.