I have also been a bit unclear about what the references to the "red button" mean.
In terms of danger to other trains, the time taken to stop the train wouldn't really have made much difference as far as I can see. As soon as the front had started moving onto the crossover, there would be a danger to anything coming the other way, and it wouldn't have been removed even with an instantaneous reaction from the driver.
But if there's a red button that sends a message to stop other traffic then clearly, the sooner it's pressed, the greater the likelihood of it preventing something.
RAIB reports are usually very thorough and they quite often go into some detail on aspects of what happened, even if they weren't "causal" or had no impact on potential outcomes - with some explanation of why the report decides they need not be considered further.
Also, sometimes they'll contain some discussion of something that happened, decide it had no impact in this case but nonetheless include a recommendation that's relevant to it, because it's been identified that it could be significant in other scenarios.
I read these reports as a complete layperson (and am always impressed with how well written they are) but this one feels like there's a part of what happened that has simply escaped discussion.
None of this needs to mean there's any criticism of the driver ... to me it's entirely understandable that it would take some time to realise that something's not right and react to it. But I might expect some discussion of things that could be changed to make it more immediately obvious to a driver that a wrong route has been taken.
Alternatively, perhaps the answer is that it doesn't actually matter too much because the system as a whole did virtually instantly detect something was wrong (there's a mention of multiple alarms going off in the signalbox) and the signaller had the means to stop other traffic. But it's not discussed.