When can you swap the words and remove "of" like "the cafe of the uni" vs "the uni cafe"? Which one is more natural for everyday English?
This is an example only and I'm looking for a general answer. Noun1 + of + noun 2 vs noun2 + noun 1
Colloquially, "uni cafe" is preferred.
However, if there were a conversation, such as,
"Do you prefer the cafe at the hospital or the university?"
"The cafe of the uni!" could be an answer (unless the hospital served better food), emphasizing the choice by the phrasing.
Caveat: NGRAM finds the phrase with the preposition almost nonexistent. ("uni cafe" was not found, so "school cafeteria" was substituted) .
A university might have more than one eating place attached to it, so let's consider 'library'. On campus, we would normally talk about the University Library, but (before I retired) I might have said to someone I met when away from home "I work in the library of the University of Derby".
I don't think it's possible to formulate a general rule, but mostly we would use the shorter form when the context is obvious (to use Dr Moishe's example, table leg when sitting at a wonky table), and we would be more likely to use 'the X of the Y' when it's necessary to explain the situation. (I'm looking for something to put under the leg of the table in my room, because it's shorter than the others.)
The most common phrasing would be neither. Usually, we would say "the uni's cafe".
"The Uni Cafe" could work as a business name. It can also be used in a sentence where it is interchangeable with an actual name.
"How about we meet for lunch tomorrow at the uni cafe."
"The Cafe of the Uni" is weird because "uni" is colloquial and the use of "of" is more formal.
"The cafe of the university" puts a bit more emphasis on "the cafe". It is a bit like saying...
"The university's cafe, on the other hand, is an independently owned and operated business."
... but without italicizing "cafe".
"The cafe of the university, on the other hand, is an independently owned and operated business."
This sentence structure is also useful when you want to decorate "the university" with lots of adjectives.
For example...
"The cafe of the Frank Lloyd Wright Metropolitan University of Neoliberal Modern Architecture features tensegrity inspired furniture."
(1) Noun1 + of Noun2 or
(2) Noun2 + Noun1 (the attributive noun form, not the possessive 's form mentioned in OP's title)
For simplicity, I will use the 2 numbers in my descriptions below.
I prefer (2) for its conciseness if Noun2 is not too long. For a long Noun2, the Noun2 + Noun1 phrase becomes unclear; readers will start to wonder where the actual noun is. Using (1) would improve the sentence in such a situation.
Thoughtco Usage Guideline: Multiple Attributive Nouns further explains this problem caused by noun clusters and gives an example:
FAX TRANSMISSION NETWORK ACCESS COST OPTIMIZATION PROPOSAL
The problem arises when a whole slew of nouns are crammed together. The poor reader's brain has no way to decode this mess until he or she has already gone through it once.
That FAX example to me may not be so bad if the long attributive noun is in the title as readers should still be able to find the actual noun quite easily. However, if the long string is not in the title (as a warning) and appears suddenly, surrounded by more words in the document body, it would affect the readability.
To answer your questions on when we can use (2), and when only the (1) is acceptable, I would say there are no fixed rules against using (2). Thoughtco says
it has always been legal in English to use one noun to modify another noun
and I extend the application to reasonable noun strings.
Normally, I would use this construction if the particular Noun2 + Noun1 phrase is generally accepted and does not affect the rest of the sentence. In the correctly phrased
the leg of the table in my room
Noun2 has a post-modifier, and hence (2)
the table leg in my room
would be less clear than (1).
For the cafe question, my preference is the university cafe.
