First, thanks for providing a really nice explanation of the derivations of the name in the British Isles. However, I would like to correct some misconceptions regarding Adams's of German descent. There were and are to this very day a number of people in Germany with this surname. One merely needs to check the LDS records to verify this. It is thought that usage of the surname there goes back to conversion of the German tribes to Christianity. My own family line originated in the Eifel region of the Rheinland near the French border. So far, we have been able to trace our line back to ~1730, not exactly ancient in genealogical terms but definately predating emigration to America ~ca 1847. There was no need to "anglicize" a surname that was identically spelled in the original German.
Granted, in our family first names became anglicized. "Nicolaus" became "Nicholas"; "Bernhard" - "Bernard", or even "Ben". But the idea of families holding this surname for hundreds of years suddenly deciding to change it (from "Adams" to "Adams") to blend in during WWI or WWII is, well, not accurate.