Whilst I may not be of assistance to you directly, I have a small knowledge of the name, its current disposition in the uk, and the lineage back through my own father, Archibald Henry Tappin, who was born in Islington, (N. London) in 1916 and died in Edgware, Middlesex in 1982.
(You will notice that I have a 'G' on the end of my surname, on which hangs a long tale!)
The name Tappin is of Saxon origin, being brought into southern England between 5th. and 9th. century. Most books regard its original meaning as referring to 'a tribal retainer' or 'village warrior'.
I have found that the name is most prevelant in the form of Tappin or Tapping to the west of London, in the counties of Buckinghamshire and Oxfordshire, which is where (apparently) the most nrotherly extent of Saxon England.
However, I do know that there were Tappins, two brothers, who were leading lights in the anti-slavery campaign immediately prior to the American Civil War.
Perhaps this fact had a bearing on the choise of family names for your forefathers?