It’s a good podcast. Well worth a listen. (Andy Orchard is my dissertation advisor, btw.)
As to its receiving little attention until the 19th century, Anglo-Saxon and Germanic studies and philology weren’t really subjects of study until then. Very few people from the late Middle English period through to the nineteenth century could read Old English. (There are probably more people alive today who can read Old English than the number who lived from 1400–1800.) Also, the poem itself had been miscatalogued. It had been listed as a history of Danish kings, probably by someone who read only the opening lines.
Thorkelin, who discovered the poem in 1796, was the national archivist of Denmark, and he came to the British Museum to copy the poem, and other documents, for the Danish archives. He realized that it was a significant work of literature and hired an amanuensis to copy it. He also made a copy himself. (Ironically, the amanuensis’s copy was a better one, probably because the man didn’t read Old English and just copied what he saw without trying to read and interpret it.) The two “Thorkelin” transcripts were later invaluable in reconstructing the portions of the poem damaged in the Ashburnham House fire of 1731—the manuscript deteriorated significantly after people started handling and reading it; Thorkelin and his amanuensis saw it in a much better condition. Thorkelin’s first attempt to publish a copy of the poem was destroyed in the Battle of Copenhagen in 1807. He finally published the first print version and translation, in Latin, in 1815. Following that, the study of the poem took off.