Contemporary Druidry is not a continuation of the ancient druids, and the reason is simple: the ancient druids left nothing to continue. They wrote nothing down, and what Greek and Roman authors reported about them is thin, second-hand and shaped by conquest. There is no route by which their teaching could have reached the eighteenth century.
The actual sources of modern Druidry are traceable and mostly recent. Antiquarian speculation supplied the setting, with Aubrey and Stukeley wrongly assigning the great stone monuments to druids; Stonehenge in fact predates any attested druid by well over a thousand years. Freemasonry supplied the lodge structure and the graded curriculum. Romanticism supplied the tone, and an appetite in Wales, Ireland and Scotland for an indigenous heritage supplied the audience. Iolo Morganwg supplied a great deal of specific content, and much of what he supplied he had invented, including ritual material and triads he claimed merely to have transcribed.
The twentieth century added what makes it a religion rather than a club: the eight-festival year settled in the 1950s in conversation with Gerald Gardner, a nature-centred theology, and from the 1970s a serious engagement with Irish and Welsh medieval literature and with archaeology.
Ronald Hutton set all this out at length in Blood and Mistletoe (2009), now the standard history. The movement absorbed his findings with little resistance. Druid orders today generally describe themselves as a modern spirituality inspired by the ancient druids rather than descended from them, and several publish their own frank accounts of the Iolo problem.