The (US) National Archives does have a guide to Citing Records in the National Archives of the United States (pdf). For paper documents, they expect you to mention which regional building it is in, and for electronic documents it should be "National Archives and Records Administration" (NARA).
Having said that, they call themselves "US National Archives" (or USNatArchives) on Twitter, Flickr, Facebook, Tumblr, Pinterest, Github and so on. So "US National Archives" (rather than National Archives US) would be an acceptable short form.
Although the UK does use "UKNatArchives" as an account name, they mainly refer to themselves as The National Archives of the UK (capital on The is important).
So to make the difference clear, I would use (words and abbreviation):
- "US National Archives (NARA)"
- "The National Archives of the UK (TNA)" (or maybe "The National Archives UK")
As Adrian mentioned in a comment, the archives in other English-speaking countries almost all describe which nation they are for:
- "Archives New Zealand"
- "National Archives of India"
- "Library and Archives Canada"
- "National Archives of Ireland"
- "National Archives of Australia"
- "National Archives of Nigeria"
- "Kenya National Archives and Documentation Services"
- "The Jamaica Archives and Records Department"
- "National Archives of Fiji"
- "National Records of Scotland"
However, South Africa has the "National Archives and Records Service (NARS)" often with no country mentioned, so should have "of South Africa" added to avoid confusion with the US one.