The text on the memorial says that the original tree was burnt by the people at the Reform League meeting at the tree on that day in 1866 but we can find no confirmation of that. Wikipedia has quite a lot to say about the meeting but no mention of the tree being burnt and, apart from the 'railings affair' when the closed Hyde Park was stormed to gain entry, the estimated 200,000 people there seem to have been peaceful.
However we did find this reminiscence of the tree being burnt, in 1875, from Select Narratives, Annual Events, etc., during twenty years' Police Service in Hyde Park by ex-sergt. Edward Owen: "… the 'original' stood; I believe an elm like its neighbours, but not a vestige of green or anything to indicate that species simply a stark, blasted-looking old trunk, dead as a doornail, whether from lightning or old age, it had fallen into such a state, I am unable to say, but that is how it appeared in the year 1875, and was recognised as "The Old Reform Tree." The occasion of its demolition, or the cause of it, happened at a meeting or demonstration in the summer of the year mentioned above. It was not a political meeting, but a trade grievance, and I remember very largely attended. So far as the meeting was concerned it had gone off orderly and quiet, resolutions had been passed, and people were really dispersing homewards. I may add it was on a week-day, and took place in the evening, I presume to give employees every facility to attend ; however, it was getting dusk, when suddenly smoke and sparks were seen issuing from the old tree, and it became apparent it had been set on fire, and that we conjectured, by mischievous boys ; burn and smoke it did alarmingly, for it was nothing more than a lump of tinder…... We could do nothing without water to put the fire out, …… We cleared the crowd back some twenty yards from the smouldering tree until the arrival of a small manual fire engine, brought by a couple of firemen ; ….. arrived and soon put an end to any sign of fire, and the crowd finally dispersed. To prevent a repetition of a similar scene, the Park authorities soon decided to have it removed altogether. Still there is the space where the old tree stood, if any of my readers care to take a walk and see as I have described."
We conjecture that either the people who worded the plaque have conflated the two meetings into one, or that the tree suffered from flames twice.
The tradition of free speech at Speakers Corner (at the north-east corner of the Park) grew out of the tradition of political meetings at the Reform Tree. It's not clear when that tradition started, nor when the tree got its moniker. It's likely the name originates from the 1866 meeting.
The picture shows the meeting at the Reformers Tree, 29 June 1866. The huge horseman in the background can only be the Duke of Wellington at Hyde Park Corner. And the tree in the picture, to us, looks more like the elm remembered by Sergeant Owen rather than the oak as described on the memorial.