As far as I understand Times New Roman is not a free font but a license to use it for personal purposes is included in products like Microsoft Word. Suppose I do not own a copy/license to any such product that includes the font. Can I legally just download the font from the internet and use it to write and publish an academic paper?
The question of whether publishing a document that uses a font is a violation of the copyright of that font depends on the method used for publication, but leaving that aside, in the specific case of the font Times New Roman, the answer is a qualified yes.
Times New Roman was one of the fonts included in Microsoft's Core Fonts for the Web package in the late 1990s. This was a package of specific fonts which Microsoft made available for free distribution. The license that was used specifically allowed redistribution of the fonts as long as they remained in their original form and were not sold or bundled for profit.
Microsoft discontinued the project a few years later, and newer versions of the fonts are no longer available under the same license, however the original font license remains in effect for the specific versions which were released under it.
This version of Times New Roman is quite old and has a limited character set, but it's still usable. As long as you distribute the original font package unchanged with the EULA intact, and do not sell it, you are allowed to distribute an unlimited number of copies of the font.
It is not legal to copy fonts that happen to be findable on the internet but not licensed for your use: that is copyright infringement. Certain fonts may be automatically sub-licensed to you via the operating system, but let's say that you found a font on a pirate site and downloaded it. That is copyright infringement, which is illegal. Personal-use infringement often receives the benefit of the blind eye so you may be able to "get away with it".
"Publication" is a very broad and non-legal term. You can submit a manuscript created with a pirated font, and a professional publishing house will substitute licensed fonts. But these days, many publications are self-publications, so it is possible (depending on format) but rare to actually embed a font in a document. That would be infringement as well, and more likely to get the attention of the copyright holder. It would definitely be infringement to upload the pirated fonts along with the document and encourage people to copy those fonts illegally along with the document.