@CloudyMrs As a Scot living in the US for over 25 years ... Americans do often say "New Year's" but it's shorthand for "New Year's Eve (or Day)." It's a possessive, not a plural.
Some people will write "New Years," but that's a failure to include the possessive apostrophe, which I believe happens sometimes in Blighty as well.
The thing that puzzles me is the difference in spoken emphasis. Americans say "Happy NEW Year" while in Scotland I think we always said "Happy New YEAR."