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Bardsley’s Dictionary of English & Welsh Surnames describes the name Gott as a topographical name referring to ‘a water channel or drain as per goyt’, as in ‘at the gott’ (‘atte Gott’ ) . Earliest reference quoted is in 1315 of John de la Gote in Yorkshire. He does not cover GOTTS.
Reaney produced a major work on surnames: Dictionary of English Surnames. He suggests the origins of Gott and Gots are petname forms of Gocelin (pronounced Gotselin), with references to records in 1319. This contradicts Bardsley who has Gocelin as sounding like Jocelin under the definition of GOSS, so does the name Gocelin turn into hard and soft variants? He also references:
In Yorkshire & Lincolnshire he suggests the name could be of Breton origin, but that Godui Gott is too early for this influence, and could be from a second origin deriving from the Middle English words gotte, gut meaning gut, guts, a corpulent or greedy person.
MA Lower’s Dictionary of Family Names of the United Kingdom (1860) has GOT as a baptismal name,with Wulfrici Gott as the example, but no GOTTS.
Hanks & Hodges Dictionary of Surnames propose Gott derives from Norman English & German, a shortform of Germanic compound names with the first element of God. Their alternative is as Jewish Ashkenazy from modern German, but this is all in relation to compound names such as Gottschalk. Gotts is not mentioned. (Neither do they provide any references, so they may be thinking purely from a German perspective.)
The Oxford Dictionary of Family Names in Britain and Ireland has a number of possibilities:
It includes the references quoted above under Reaney, and also: