Both Sassoon and Owen had their poetry published in collections during the war (1917). Sassoon became more popular for his anti-war stance and content in the late twenties-early thirties, and Owen entered into his eponymous position today during the sixties, largely for the same reasons as Sassoon during the thirties.

Todman and Bond have written extensively on this subject, and /u/NMW has some pretty decent answers to questions just like this one.

As /u/DuxBelisarius below has noted, I've had cause to cover a similar question before -- you may find it useful.

In the meantime, Dux's timeframe is fairly accurate. While Owen and Rosenberg remained fairly obscure for decades, poets like Sassoon and Graves were well known even during the war itself and capitalized on this fame in its aftermath. 1922 saw the publication of C.E. Montague's Disenchantment, a novel-essay that paved the way for the great "War Books Boom" of 1927-33. It was during that period that the kind of viewpoints that have become so dominant about the war were given their most exuberant voices; the period saw the publication of Hemingway's Farewell to Arms, Remarque's All Quiet, Graves' Good-Bye to All That, Sassoon's Memoirs of an Infantry Officer, Blunden's Undertones of War, Brittain's Testament of Youth, etc., and such works -- along with the "disillusioned" poetry that so echoed their sentiments -- swept the field.

Owen and Rosenberg's rise to popularity in the 1960s was due as much to rediscovery as to repopularization -- on Owen's part, at least -- by Sir Benjamin Britten. His War Requiem took Owen's poetry as the substantive basis of its text, and the young poet's songs of protest found many willing ears in the ferment of the rising 1960s counter-culture. They've attained a supremacy rare in any field, and only slightly eroded in recent years by attempts to examine other areas of the canon (women's poetry, civilian poetry generally, "patriotic" poetry, etc.).

Anyway, more details on all of that are available at the link above!