This is specific to the jurisdiction in question.
In the UK, "The Times" is unambiguously understood to refer to the newspaper published in London; hence, the usage of "Times" by another newspaper in the UK could be considered confusingly similar and therefore a trademark infringement.
On the other hand, in the United States, "Times" is used by several different newspapers, among which are The New York Times and The Los Angeles Times. While I am not a lawyer and you should consult with a proper attorney for actual legal advice, using "Times" as part of the name of a newspaper is unlikely to be considered trademark infringement unless your paper bears similarities to other papers of a kind that could lead to confusion.