Anglican parish registers exist from 1586. As of late 2004, the registers have not been deposited in the Lincs. Archives, nor have they been filmed by the LDS Family History Library. And copies of the Bishop's Transcripts have not been deposited either.
The Lincolnshire FHS has published several Marriage indexes and a Burial index for the Louthesk Deanery to make your search easier.
Kelly's 1900 Directory of Lincolnshire places the parish in the rural deanery of Calcewaith No. 1.
The name Burwell is from the Old English burh+wella meaning "spring by the fort". In the 1086 Domesday book, the village name is given as Burwelle. A. D. Mills, "A Dictionary of English Place-Names," Oxford University Press, 1991.