The books were all released within months of each other. Tolkien wrote what he perceived as one book: The Lord of the Rings. The publisher decided the book was too long and broke it up into 3 volumes. But they had all been completed before the first was published in July 1954. The second volume was released in November 1954, and the third in October 1955. So there wasn't a huge amount of speculating and such--there wasn't time. Also, The Lord of the Rings collectively was only Tolkien's second book, the first having been The Hobbit of course, which was a children's book and published 17 years before The Fellowship of the Ring. George R.R. Martin's career has been doing a slow burn for decades, while Tolkien's was kind of isolated. So there wasn't this huge fandom he'd already developed. While Martin's work is huge now, there was little attention or fanfare when the first book of the series, A Game of Thrones, was published in 1996. Also, the book wasn't published in the US for quite some time afterwards, so there wasn't any huge Tolkien craze in the US, even though there were science fiction conventions, and a ready audience who would likely have gone nuts for the books had they been able to get ahold of them in the 1950s. So the popularity of the books grew over time. Fandom in the US for Tolkien didn't go nuts until the 1960s, when some pirated versions came here, and then it was published here in paperback, finally. F8or example, it wasn't until 1969 that a parody of Lord of the Rings was published, called Bored of the Rings, by the Harvard Lampoon.

As for fan activities: people did all the things you're mentioning, about drawing the characters, writing fan-fiction, and discussing Tolkien, once his work became better known. These sorts of activities were all centered around fan conventions--not specifically LotR conventions, but general science fiction conventions. Such activities existed for other authors and genres also at the conventions. People wrote music to accompany the songs in LotR and had singing contests, too. But pre-internet, these activities stayed outside of the mainstream and were the purview of the committed fans who spent time and money attending conventions that were sometimes far away. People who wrote fan fiction had to mimeograph (and later xerox) their work and give it away or sell it at conventions. Or they'd have ads in the back of pertinent magazines and sell them there. It was a slow process that involved a lot more time and commitment.

What is perhaps a better model is the craze for Dickens books. He published many of his novels in parts over many many months. He had a huge following and people were extremely eager to find out what happened next with each installment. Allegedly, people met the ships from England, hoping to get the next installment as soon as possible, and there was wild speculation, and fan distress if a favorite was killed off, that sort of thing.

As /u/bettinafairchild has said, LOTR fandom has existed for a very long time, and pre-internet, was very similar to offline fandom interactions now -- zines and conventions.

The fandom did grow proportionally to the availability of the books -- that is, it was relatively small until the publication of the second edition of the hardcover and the paperback edition.

LOTR, in particular, became a cultural touchstone for the counterculture in 1960s America. (To the point where people argued that Sauron was an analogue for the US military draft... considering Tolkien was not particularly anti-war, nor countercultural, this interpretation involves a rather liberal amount of saying "the author is dead" and handwaving.)

College students would wear "Frodo Lives" or "Go Go Gandalf" buttons (source behind paywall). By 1966, the University of Wisconsin Tolkien Society had a journal. The UW Tolkien society has met almost every month since its founding, almost 50 years ago.

The Mythopoeic Society was started as a Tolkien discussion society, and the mostly-New-York-based Tolkien Society of America had over 1000 members in 1967. By 1969, the books were popular enough to produce a parody spin-off, Bored of the Rings. The same year, the first Tolkien conference was held. This NYTimes article describes an early Tolkien society.

Since Tolkien was an academic and it very much shows in his writings, the fandom produced significant literary and critical analysis of the books. Here's a 1992 list of books dealing with Tolkien's works. That is not to say that all fanworks were academic -- fanfiction appeared in zines, though it would become much more widespread once the Internet existed -- rec.arts.books.tolkien was created in 1993.

Most Tolkien fans, obviously, did not participate in written discussions of his work or in-person conventions, but the list of works inspired by Tolkien is spectacularly long (and includes, among other things, songs by Led Zeppelin). LOTR essentially created a genre of fantasy -- inspiring, among others, Marion Zimmer Bradley, David Eddings, and Terry Brooks. In a very real sense, ASOIAF would not exist without Tolkien, as it spawned much of the high fantasy genre. ASOIAF does not adhere to many of the genre norms, but much modern fantasy is, at least in part, a reaction (and occasionally backlash to) the genre as set out in LOTR. (See also China Mieville's writing on the subject.)

LOTR fandom is quite well-documented -- for more information, the Wikipedia article is a good place to start. Fanlore also has information. The wikipedia article on Tolkien's influence discusses Tolkien's influence on other writers.