Thinking about why it is that movements with such a comprehensive liberal program could progress and prevail in the 19th century but not the 17th, I suspect that it is a matter of the degree to which the plebeian classes who would obviously be the backbone of the Leveller program had bought in to capitalism as the prevailing normality, and the degree to which the monied upper classes perceived they had done so.
In the 19th century such a radical political program proved (although doubters among the ruling classes were many, and their resistance died hard) compatible with the ongoing power and privilege of wealth, and the continued capitalist organization of the economy which meant the nations revolutionized thus politically remained in the mainstream of the Euro-Atlantic-Global economy. This meant both that powerful toes were not trodden on too hard and that the industrial strength of these nations continued to grow, to remain competitive with their less progressive rivals and also integrated with them by trade.
Now look at the Levellers of England, look at their very name! (I don't know to what extent that was a name they gave themselves, and to what degree it was an insult by their enemies that they turned around to adopt with pride). It might be, that if the monied powers of England had only given them a chance, that they too would prove like their more numerous spiritual descendants some centuries hence quite willing after all to leave great wealth in the hands of the greatly wealthy; to hold that factory working conditions were the outcome of a free bargain between a master and free workers, to regard capitalism as the legitimate and natural outcome of "the pursuit of happiness" as Jefferson euphemistically put it. Perhaps. But does it seem likely? Whoever called them "Levellers" apparently didn't think so!
The Levellers, in context, are not so much proto-liberals as proto-socialists. It is probably valid and certainly interesting to speculate on whether they could have seized power, but if they did it would be after a tougher fight than their latter-day apparent echoes had in the 19th and 20th centuries, and the outcome would be quite different, probably, than something like the French Third Republic. I imagine that these Levellers would, correctly, regard the concentrated power of wealth as one and the same with the concentrated power of aristocratic government they were openly attacking. And whether or not their leaders could be persuaded to let concentrations of private wealth alone, the owners of that wealth would not want to take any chances and would fight the Levellers, with the considerable resources they had. If the Levellers somehow won, it would be over the smoking ruins of the workshops and rented farms and (overseas) plantations of these magnates, and whatever open-mindedness individual Levellers may have started with about the rich, they'd have changed and closed those minds long ago in the bitter struggle. They are not going to see capitalism reinstated; parallel to their ideas of political equality they will believe in economic equality as well, living up to their name in redistributing the wealth of English landholdings and the franchise for urban craftwork evenly among the working people.
Now if they do that in a simple and straightforward way, they will create a peasant Utopia for one generation, that will start to come apart at the seams before the next has fully matured. Some farmers (in the American, not British, sense) will be more successful than others, and so will some craftsmen. Should they be allowed to use their wealth to transform the manner in which work is done, or must they continue to work as their neighbors do? All around them in Europe, great fortunes are made by using capital to hire workers to produce more efficiently and under control of their wage-paying master; if this path is forbidden by political vigilance by the Levellers (and not allowed in disguise by a corruptible regime) can the democratic English find another path to reorganizing labor so as to produce more efficiently (but unlike under capitalism, for the good of the community as a whole)? Or will England slip behind, becoming poorer per capita (the more so because the buccaneering capitalists of London have been preempted from muscling into the East India trade and the slave plantations of the West Indies) and the nation of England weaker militarily for lack of revenues and resources? Can the Levellers have not only the political and moral but technological vision to develop industrial technology on the basis of democratic fairness, or must the power that capitalism brought to the holders of capital always come at the price of mass exploitation and immiserization?
These questions did not arise in the 19th century because by then the potential of a nominally liberal, notionally equal populace to nevertheless participate in the capitalist game without too much quibbling and quarreling had been demonstrated. It was widely known that a nation could profess such sweeping theories of equality and yet host the full spectrum of class stratification that capitalism assumes and runs under. Well actually I'd say they sort of did arise, in that extreme reactionaries were able to point to more radical ideologies that arose in response to the injustices of a nominally liberal order, and that therefore the larger mob who today might be pleased to have a vote and abolition of invidious class degradations, and then placidly show up to work in the fields as hired hands or in factories as workers, might tomorrow decide they had been fooled and they ought to be upping the stakes. Resistance to even basic liberal rights was based on the fear that in the end it was only open class repression that kept the social order in shape, and each concession just made the task of the inevitably necessary mass disciplinary actions that much harder.
But nominally liberal societies were the most powerful in the 19th and early 20th century, and this was no accident, so despite the misgivings of reactionaries, the "Leveller" agenda was the norm then.
I don't think the Levellers of the 17th century had learned to make the accommodation with their monied betters they needed to do, so the problem of their takeover and the style with which they'd have run things seems entirely a different question than what Victorian era populaces actually did with their power.