Verbatim usage of content is copyright infringement obviously but speaking English is not. Learning from content is not copyright infringement either. I don't know if NYT has a clause for this type of usage of their content but still I don't think it would be covered by copyright the way I understand it
As long as an LLM rephrases what it learned and not regurgitate verbatim text, it should be fine but we'll see what the judge says