Fan Fiction
As a comment mentions, "fan fiction" is not a legal concept, and has no particular protections or rights. Most fan-fiction is in fact a copyright infringement. Some authors and publishers choose not to take legal action against creators and distributors of fan fiction, perhaps on the theory that it serves to advertise the original works, and helps build good will. Others attempt to prevent the publication and distribution of fan fiction, and may bring suits in some cases. Many have found it impractical to respond to every piece of fan fiction. That does not make such works legal, or ensure that any given author will not take legal action against a given work of fan fiction.
A few authors have explicitly permitted works of fan-fiction, either specific instances or in general. That is not the majority position.
In any case, the kind of faux magazines described in the question are not fan fiction as that term is usually used. Nor do they seem to be parodies as that term is used in copyright law. They may or may not be unlawful.
IP Rights
There are two main kinds of IP that might be involved here: trade marks and copyrights.
Trade Marks
Under the UK Trade Marks Act of 1994 (section 10 (1))
a person infringes a registered trade mark if he uses in the course of trade a sign which is identical with the trade mark in relation to goods or services which are identical with those for which it is registered.
Paragraphs (2) and (3) of section 10 of the Act also contain the qualifying phrase "in the course of trade". Those are the paragraphs describing infringing actions under the Act.
It would appear that no use of a mark that is not used in connection with good or services that are being sold, rented or leased, offered for sale, rental, or lease, or used in connection with advertising such goods or services is a "use in trade". The uses of trademarks or imitations or parodies of trademarks described in the question do not seem to be used in trade. Therefore, use as described in the question would not seem to infringe any actual trade marks under UK law. Much the same would be true under US law, and, to the best of my understanding, under the laws of most other countries.
Copyright
Under The Berne Convention, and under both UK and US copyright law, a work that is based on another work is a derivative work This include translations, imitations, parodies, and sequels.
Under US law there is a limited exception for parodies which comment on the original, but not for parodies which use the form of the original to comment on other issues, or for general humor; those are considered satire, and do not fall within the exception, which is an aspect of fair use. In any case that does not apply in UK law.
Whether a work is so closely based on another that it constitutes a derivative work is a judgement call. Simply being an instance of the same general plot such as "boy meets girl" or "the man who learned better" is not enough. A detailed chapter-by-chapter or paragraph-by-paragraph similarity would generally make the newer work a derivative work.
If the connection between source and later work is not easy to notice, it is less likely to be treated as a derivative work.
If a work is derivative, and the source work is still under copyright protection, permission is required. Otherwise it is a copyright infringement to create or distribute the derivative work.
Place names, words and phrases, and facts
Place names, real or invented, are not protected by copyright. Their use is not infringement.
Individual words and short phrases, such as book or song titles, are not protected by copyright. Their use is not infringement.
Facts are never protected by copyright. However, the selection and arrangement of facts may be protected, if there is some creativity or originality in the design. An obvious or natural arrangement, such as alphabetical, numerical, or chronological order, will not gain copyright protection.
The specific words used to express a fact may be subject to copyright protection, but not if they are the only way, or the only natural way, to express the fact.