My guess is that the court will likely find in the Times favor, because the legal system won't be able to understand how training works and because people are "scared" of AI. To me, reading a book, putting it in some storage system, and then recalling it to form future thoughts is fair use. It's what we all do all the time, and I think that's exactly what training is. I might say something like "I, for one, welcome our new LLM overlords". Am I infringing the copyright of The Simpsons? No.
I am guessing some technicality like a terms-of-use violation of the website (avoidable if you go to the library and type in back issues of the Times), or storing the text between training sessions is what will do OpenAI in here. The legal system has never been particularly comfortable with how computers work; for example, the only reason EULAs work is because you "copy" software when your OS reads the program off of disk into memory (and from memory into cache, and from cache into registers). That would be copyright infringement according to courts, so you have to agree to a license to get that permission.
I think the precedent on copyright law is way off base, granting too much power to authors and too little to user. But because it's so favorable towards "rightsholders", I expect the Times to prevail here.