My father was Ken Meacheam (died 2006). Both he and his only sibling, brother John, were good local historians and contributed to local history in Dannevirke. Their parents were John and Rosas' son William (Bill) and Rose Meacheam (a cousin, but not a blood relative, adopted, her mother was a domestic for the Comptons, Marquis of Northampton).
My time on-line is limited at the mo., but John Meacheam, born 1848, may have been the son of William Meacham (different end of name), and Mary. In the 1861 Census they are recorded, William's birthplace was Newton Longville, Mary from Bletchley, Bucks. In notes from Dad I read last night, he recalls a bible, in the front of which was written "1858, gift from the Newton Longville Baptist Church, to John Meacham". The 'eam' end to our name might be the product of John varying it.
Hope this helps. I have stacks else I might convey from notes Dad made and left with me.