This is a great question - but it is also terribly difficult to answer. There has long been a perception that "our current time" (whenever that is) is less superstitious and gullible/believing than the previous generations. This perception is documented in Chaucer, for example. The following is an excerpt from my recent The Folklore of Cornwall: The Oral Tradition of a Celtic Nation (2018):

There is evidence that people have always thought their beliefs in the supernatural were fading and that earlier generations were more fervent in their fairy faith. Asserting that a belief in these entities was a bygone facet of English heritage features in Chaucer’s fourteenth-century introduction to ‘The Wife of Bath’s Tale’, which the character sets ‘In the olden days of King Arthur [when] … all this land was filled with faerie’. The Wife of Bath adds, ‘This was the old belief’. It is a theme that appears to have resonated over the centuries with a repeated assertion that people regarded those from previous centuries to have possessed a stronger faith in the existence of a fairy world. Writing in 1997, Linda-May Ballard cites Jeremiah Curtin as describing the idea of a waning belief in the fairies in his 1895 publication on Irish folklore. Ballard then poses the question, ‘Might it be that the idea that fairy belief is fading and belongs to the past, is part’ of the wider tradition embracing the belief in these supernatural beings? (sources: John H. Fisher, editor, The Complete Poetry and Prose of Geoffrey Chaucer (New York: Holt, Rinehardt and Winston, 1977), p. 120; Linda-May Ballard, ‘Fairies and the Supernatural on Reachrai’, in Narváez, The Good People, p. 91; note 9)

Although not specifically from Cornwall, Seeing Fairies: From the Lost Archives of the Fairy Investigation Society, Authentic Reports of Fairies in Modern Times, provides evidence of British tradition enduring into at least the mid-twentieth century. Modernism affected but did not extinguish fairy traditions. A Cornish example from 2017 reinforces the idea that while folklore may change, aspects of belief can defy intuition by lingering over time. The Packet, a newspaper serving Falmouth and Penryn in Cornwall, reported the one-hundredth birthday of Falmouth native Molly Tidmarsh. The centenarian implied that some of her good fortune in living so long may have been due to her birth under a ‘piskie ball’, a round lump of clay, fired together with one of the tiles used on the roof ridgeline of her family’s home and business. Molly suggested that these objects were created to distract piskies who sought to come down the chimney to cause mischief for the occupants of the house. Instead, the piskie ball would entrance them, and they would dance around it until dawn, at which point they would disappear. It is unclear, and largely unimportant, if Molly Tidmarsh believed good luck was hers because she was born under the ball; what matters here is that piskies featured in a newspaper article in 2017 without a need to explain what they were. Molly remembered a tradition of the early twentieth century and it still resonated with readers one hundred years later. (sources: Marjorie T. Johnson, Seeing Fairies: From the Lost Archives of the Fairy Investigation Society, Authentic Reports of Fairies in Modern Times (San Antonio, Texas: Anomalist Books, 2014); The Packet, 22 August 2017).

This is a limited amount of evidence, but it points to something that is an irresistible truth about folklore: everyone has it and it is not something that is shed with any "sophistication" of modernism.

We can attack this question in another way. Traditional, pre-modern European folklore included folktales, which were generally recognized as fictional. These included fantastic creatures taken from belief, but the stories themselves were fictional - so if your question refers to something like "Jake and the Beanstalk," for example, that story was normally told as fiction even in its traditional, pre-modern setting. What you are referring to are the legends, the narratives generally told to be believed. Despite what I have noted, belief in many traditional creatures - "the dragons and boggarts" of your question, admittedly declined with growing urbanization in the nineteenth and early twentieth century. Because belief persisted and yet people did not necessarily want to admit it, I wouldn't trust a poll even if it had been taken on the subject. While conducting research in Ireland, 1981-1982, I spoke with a young woman from Kerry who said that she believed in the fairies when she was home, but that she found the belief fall away when attending university in Dublin - only to have the belief resurrect when returning home. That fluidity serves as a nice expression of how folk belief can come and go.

Then there is the problem of which folklore is the subject of the question. The older traditions of Britain may have faded - or are in the process of fading, but new ones are taking their place. While enthusiasm for ghosts, for example, persists, new beliefs are just as easily taking hold. There is a modern growth of traditions surrounding angels, but then there are a full range of entities and beliefs associated with urban legends. The subjects of beliefs mutate and some may drop off, but because folklore is ubiquitous, belief persists.

In short, there is no set time when disbelief became prevalent although it has always been believed to be "the previous generation." One can argue that the time of disbelief has yet to be attained (and likely never will) since every generation attaches itself to some belief structure.