In the present day, when an author like J. K. Rowling uses initials in the author's name on books, it is plausible that this was an explicit choice / conscious decision, and the author had a special reason for doing so. (In Rowling's case, the reason was that the publishers thought young boys might not want to read a book written by a woman.)
This plausibility is based on certain assumptions about "normal" names in the English-speaking world:
- A person's "real" or "proper" name, used for most purposes in real life, is in the format "GivenName Surname". (For example, Rowling's name, at the time she was writing the books, was "Joanne Rowling", and she might be referred to formally as "Ms. Joanne Rowling".)
- The name used by one's friends and family is either one's given name, or some diminutive form of it. (For example, Rowling is called "Joanne" or "Jo".)
Based on such assumptions, it is reasonable to wonder about other authors, like J. R. R. Tolkien: for example, one may speculate that he used initials because he was an academic scholar wishing to adopt a somewhat different pen name in his books for children, just as Charles Lutwidge Dodgson became "Lewis Carroll".
Most of us know, I guess, that these assumptions do not necessarily hold across different cultures. Even "obvious" assumptions may be false, and personal names around the world vary. (For instance, the children of V. S. Achuthanandan are named V. A. Arunkumar and V. V. Asha, and this is perfectly natural.) Pointers on Wikipedia: personal name, anthroponymy, personal names, personal names by culture.
But cultures vary not just across space but also across time: it is easy to forget that “The past is a foreign country; they do things differently there”.
So if one is willing to abandon one's cultural conditioning and be a little open-minded about what sort of names people are allowed to have, it turns out that:
- J. R. R. Tolkien was not doing anything unusual in using initials; it was a common practice at the time. (The fact that he was called “Ronald” by his friends and family implies nothing about his formal name: C. S. Lewis was called "Jack".)
- The name “J. R. R. Tolkien” was not a pen name that he used for non-academic works; it was simply his name.
This applies not just to Tolkien but to many early 20th-century English authors (as asked at the now-deleted question). Many authors used initials simply because many people in the late 19th / early 20th centuries used initials. We do not need to look for special literary reasons, because these authors were not doing anything special. The names with initials were not their noms de plume but simply their conventional names that they used in other aspects of life.
Let me elaborate on both these points separately.
Names with initials were common
We can take a deeper look at the milieu of E. M. Forster (born 1879), J. R. R. Tolkien (born 1892) and T. S. Eliot (born 1888), by looking at the styles of names in fields other than literature. As a proxy for getting a representative list of names, let's look at:
- Film directors: in this List of British films before 1920, there are many directors who went by initialed names, including R.W. Paul (born 1869), G.A. Smith (born 1864), A. E. Coleby (b. 1876), W. P. Kellino (b ~1874), A.V. Bramble (b 1884). Note that many of these seem to have gone by different styles of names at different periods.
- England cricket captains: A N Hornby (b 1847), W. G. Grace (b 1848), H. D. G. Leveson Gower (b 1873), C. B. Fry (b 1872), M. J. K. Smith (b 1933).
- List of Prime Ministers of the United Kingdom: H. H. Asquith (born 1852).
- Nobel laureates from the United Kingdom: J. J. Thompson (b 1856), A. V. Hill (b 1886), C. F. Powell (b 1903).
- English mathematicians: G. H. Hardy (b 1877), F. H. Jackson (b 1870), D. G. Champernowne (b 1912), I. J. Good (b 1916), W. T. Tutte (b 1917), J. W. S. Cassels (b 1922).
There are even more that did not use initials, of course. This cherry-picked list of names is only to illustrate that people using names with initials were not uncommon, and presumably there aren't necessarily special literary reasons for being part of that trend. (It does seem to be a trend: the occurrence seems to peak in that rough period.)
Note that these non-literary people did not use their initials as "pen names" for some special literary purpose — these were simply the names by which they were known in real life, the names by which articles about them are still titled.
[I picked only people from the UK, but you can also look at people in the broader British-influenced countries: we have scientists like C. V. Raman, S. N. Bose, J. C. Bose (continuing to the present day), and people from the literary world like R. K. Narayan or V. S. Naipaul.]
Doing a statistical analysis over some larger and more representative dataset of names will let you know whether authors like Forster and Tolkien were picking a name style unusual for their time.
“J. R. R. Tolkien” was not a pen name
Was the name with initials a nom de plume he adopted for non-scholarly works? No! Here are the covers of a 1922 scholarly work of his, and the second edition of another:
(Middle English, not Middle Earth!)
Tolkien first published a work of fiction in 1937 (The Hobbit). If we search on Google Books for books before 1935, and look for strings like "Mr. J. R. R. Tolkien", "Mr. John Tolkien", "Mr. Ronald Tolkien", "Mr. John Ronald Tolkien" and "Mr. John Ronald Reuel Tolkien", we find results only for the first one.
Transactions of the Yorkshire Dialect Society, 1921:

The University Bulletin, 1922

The Periodical, 1923:

Studies in English, 1927:

British Books in Print, 1920:

Take a look at these, especially the last one: do we really want to look for special literary reasons for, or even consider unusual, the fact that all these people had initials in their formal terms of address?