I don't know of any awards that Professor Tolkien won during his lifetime. It has been said the Lord of the Rings is the most important book(s) written in the twentieth century. I don't know if that is true, but I am sure it is within the top one hundred.
Tolkien won few awards during his lifetime. Awards for books were not so commonplace as they are today and, even today, awards rarely spot a classic in the making.
In April 1938, The Hobbit won a prize, awarded by the New York Herald Tribune, as the best juvenile (ie written for children) story of the season. (Letters of JRR Tolkien; letter 28.)
In 1957, The Lord of the Rings won the International Fantasy Award at the 15th World Science Fiction Convention. As a point of historical interest, this award preceded the "boom sales" of the 1960s, and led to film-maker Forrest Ackerman showing an interest in adapting the story for the screen. (Letters of JRR Tolkien; letter 202.) It is not true, as has sometimes been suggested, that the book was obscure until it was released in US paperback.
Tolkien said that he thought the rocket statuette "absurd", but the speeches at the convention "far more intelligent". He kept the statuette, which is still in the family's possession.
The Hobbit was awarded the Keith Barker Millennium Book Award Winner presented in 2000 by the Youth Libraries Group, School Library Association and Library Association Schools Library Group for the most significant children's book published between 1920 and 1939. This medal, a one-off award in memorial of librarian Keith Barker, was awarded under the aegis of CILIP, the Chartered Institute of Librarians and Information Professionals, awarders of the UK's prestigious Carnegie Medal. See www.clip.org.uk
The Hobbit did not win the Carnegie in its year of publication, losing narrowly to The Family from One End Street by Eve Garnett. Eve Garnett's book, excellent though it was at the time, is little remembered now.
The Hobbit was also named "Most Important 20th-Century Novel (for Older Readers)" in the Children's Books of the Century poll conducted by the US publication Books for Keeps.
The Silmarillion won the Locus Award for 1997. Locus is a respected US industry publication for Science Fiction and Fantasy publishing.
JRR Tolkien recorded in his essay 'English and Welsh' that " ... the only prize I ever won (there was only one other competitor) [was] the Skeat Prize for English at Exeter College ..." He spent the prize money on books about Medieval Welsh.
On the academic front, Tolkien never "took a Ph.D." was we now sometimes say - he was too busy working professionally on the kind of stuff people normally do Ph.Ds on - but he was awarded a Doctorate of Letters (D. Litt.) and Philosophy by the University of Liege in Belgium in 1954 and similarly a D. Litt by the University of Dublin in Ireland that same year. In both cases this was for his contribution to his field of philology and medieval literature in general, and his services to the universities in particular as a contributing examiner and researcher.
(This was not of course for his fiction. LotR had only just begun to be published, although he noted with some bemusement that in Belgiun he was also welcomed by the faculty as "the creator of Monsieur Bilbo Baggins", as The Hobbit had been out since 1937 and was quite well known.)
In 1972, the year before his death, J R R Tolkien was honoured as a C.B.E. (Commander of the Order of the British Empire) for his contribution to literature, and also (probably even more important to him) awarded an honorary Doctorate of Letters by Oxford University for his contribution to philology. To the end of his days, Tolkien never applied for a Ph.D., although he had done work at that level many times over, and held three Professorial chairs in his life.
In Great Britain, the title "Professor" accompanies a specific and senior academic position, or Chair, rather than a regular senior or tenured teaching post.
Many thanks to Wayne Hammond, Andrew Wells, Alan Reynolds, Nigel Evans (for expounding C.B.E. correctly for us), JRR Tolkien: A Biography and The Letters of JRR Tolkien by Humphrey Carpenter and the CILIP website www.clip.org.uk . Additional material and compilation (as most of these items) by Helen Armstrong.