(Did you have a slightly earlier version of this question that included Marie Antoinette? I was sure I saw that last night before I went to bed ...)

I'm not sure it's useful to think of pro-Louis (and pro-Marie Antoinette) sentiment as "cultlike", because it's not as though everyone in Europe shared the revolutionaries' opinions on him even at the time, outside of isolated pockets of adoration. Immediately before and during the Revolution, it was by no means certain that the king and queen of France would be executed, and it was not the unanimous opinion of even the republicans of the West that they were monsters. Gouverneur Morris reported to Thomas Jefferson, who was Secretary of State at the time, that

To a person less intimately acquainted than you are with the history of human affairs, it would seen strange that the mildest monarch who ever filled the French throne, one who is precipitated from it precisely because he would not adopt the harsh measures of his predecessors, a man whom none could charge with a criminal act, should be prosecuted as one of the most nefarious tyrants that ever disgraced the annals of human nature - that he, Louis the Sixteenth, should be prosecuted even to death.

The last years of the royal couple's (and their children's) lives had a great effect on the way that they would be perceived in the wake of their execution. In October of 1789, a mob of Parisian women invaded Versailles and threatened Marie Antoinette; the royal family was then brought to Paris and essentially imprisoned in the Tuileries Palace. The family attempted to flee in June 1791 (see my previous response on this) but were returned and placed under heavier guard. The Tuileries was itself invaded twice by the mob, and the family was moved to the Temple. The monarchy was abolished and Louis was arrested (and separated from his family) in August 1792, tried in December 1792, and executed in January 1793 following lengthy debate; Louis Charles was then taken away from his mother and sister in July, and in August Marie Antoinette was separated from her daughter and taken to the Conciergerie, where she was very ill and under constant surveillance. In October 1793, she was tried and then executed. Louis Charles was generally secluded and alone, and he became seriously ill and died in 1795. His sister, Marie-Therese, had been in the care of her aunt following Marie Antoinette's move to the Conciergerie in August 1793, but said aunt was executed in May 1794; she was told of the deaths of the rest of her family in 1795 and subsequently used in an exchange of prisoners.

This is the story of a family that was ripped apart, parents killed and children orphaned and abused at least emotionally (there are a lot of stories about Louis Charles being horrifically physically abused, but there isn't good evidence on the subject; however, at the age of 8 he was either forced to testify that his mother had sexually abused him or made to believe that it had happened, which I'd classify as abuse). Political considerations aside, there is a lot to sympathize with. The fact that Louis and Marie Antoinette both were stoic and dignified throughout their imprisonment, trials, and executions led to a certain amount of acclaim even from their enemies.

Even before the executions, Edmund Burke wrote Reflections on the Revolution in France (1790), a pro-Royalist tract that would influence other writers. Marie Antoinette's trial was published in English, along with a glowing biography; you can probably guess what stance The Cruel Massacre of the King & Queen of France took, and what kind of anecdotes it contains. The multi-volume Letters from France is similar, comparing Marie Antoinette to Mary, Queen of Scots (despite the religious differences between Mary Stuart and the eighteenth century British, she was seen as a highly romantic heroine). A French tragedy of 1794 outright called her a martyr, and an Austrian biography was very snooty about the French people for their actions.

An early positive account that was extremely comprehensive was Jeanne-Louise-Henriette Campan's Mémoires sur la vie privée de Marie Antoinette, suivis de souvenirs et anecdotes historiques sur les règnes de Louis XIV - XV, published posthumously in French and English in 1823. While not everything in it was entirely complimentary, it showed the members of the court and royal family as humans first and foremost. It also rebutted the actual charges against Marie at her trial, which she had been convicted of - draining the national treasury by overspending and sending money to her brother, the Holy Roman Emperor; leaking military plans to the counter-revolutionaries and the foreign countries France was fighting; and fomenting an internal civil war - as well as contradicting the rumors of her infidelity with Axel von Fersen and various of her close friends, and setting the record straight regarding "the affair of the necklace". Another important early historian in this vein was Alexis de Tocqueville, who published L'Ancien Régime et la Révolution in 1856. His great-grandfather had been guillotined for serving as Louis XVI's defense lawyer, and many other members of the family followed. While de Tocqueville was for the aims of the Revolution, he had personal sympathies for Louis and pointed out numerous ways that the problematic political situation was in part due to his attempts to improve things rather than keeping with the strict authoritarianism of the past.

While the historiography of the Revolution itself has generally come down on the side of the revolutionaries as part of a narrative about the struggle of the peasantry and bourgeoisie against the aristocracy and clergy, the typical view of Louis XVI has been that he made mistakes rather than that he was an essentially bad person. As a result, he tends to be popularly depicted as an incompetent ruler but a good man, with attention focused on his interactions with his family, particularly during their imprisonment, his dignity, and his love of locksmithing.