Are there any U.S. or U.K. laws that prevent academic research papers
or journals from being published?
Yes. Certain kinds of mathematical research pertaining to cryptography are classified for national defense purposes by the NSA. I suspect that other kinds of research that disclose classified defense technology could also be classified and prevented from being published.
For example, if a paper published the detailed authorization code software for U.S. nuclear missiles currently in use, that could probably be classified and prevented from being published.
In general, the restrictions would be easier to make in the U.K. which does not have a robust first amendment constitutional protection of free speech, than it would be in the U.S.
Can academics outline the methodology of illegal activities?
Yes. Describing how illegal activities are conducted generally does not violate the law.
However, this does not absolve the investigators from criminal liability if it was committed in the course of doing the research as opposed to publishing it. Also, publishing methodology of illegal activities is likely to make you a suspect when a crime is committed using those methods that you could possibility have committed.
For example, could a computer security researcher publish a paper on
unknown server vulnerabilities?