My database has many immigrants from the 1895-1940 timeframe. I filtered the 1511 individuals with SSNs to exclude the U.S.-born and the known naturalized. This left 78 who were not naturalized by 1940. Admittedly, some already had their 1st papers, and could have naturalized before their deaths. And I lack most SSN application dates.
Since the purpose of the having a SSN was to track employment and retirement benefits, early application was likely more linked to type of work (big industry) than naturalization status.