This root first appeared as ᴱ√TYALA “play” in the Qenya Lexicon of the 1910s with the derivative ᴱQ. tyalie “play, game” (QL/49). Forms in the contemporaneous Gnomish Lexicon like G. dairtha- “play” < ᴱ✶daı̯al- hint that the actual primitive form might have been ✱ᴱ√DYALA or ✱ᴱ√DAYALA (GL/29). In a table of vowel mutations added as a revision to the Gnomish Lexicon Tolkien instead had forms like ᴱ✶táliı̯èndǝ > G. teilian indicating a primitive initial t- (PE13/116); similar forms appeared in the Early Noldorin Dictionary of the 1920s (PE13/165).
In The Etymologies of the 1930s Tolkien gave ᴹ√TYAL “play” with derivatives like ᴹQ. tyal-/N. telia- “to play” and ᴹQ. tyalie/N. telien “sport, play, game” (Ety/TYAL). Tolkien continued to use Q. tyalië “mirth” in later writings (PE21/80; PE17/64), but at some point in the 1950s decided that initial palatalized dentals were no longer a feature of Primitive Elvish (PE18/93-94), making the ongoing validity of √TYAL as a CE root doubtful: see the entry on how [[at|[j] was lost after initial dentals]] in the Welsh-like branch of the Elvish languages for further details. For example, in the second version of Tengwesta Qenderinwa (TQ2) from around 1950, Tolkien indicated √TYAL was a Quenya-only root, an elaboration of √TAL:
> In Quenya initial groups ty, ny, ps, ks appeared as developments of ky, ñy, sp, sk. But being established they were used in new formations. Thus Q tyal- “play” as a variation on tal- (not kal!) (PE18/84).
Neo-Eldarin: For purposes of Neo-Eldarin, I think it is best to ignore Tolkien’s decision to remove initial palatalized dentals from Primitive Elvish in order to retain roots like √TYAL for both Quenya and (Neo) Sindarin.
In The Etymologies of the 1930s there were two roots: unstrengthened ᴹ√MER “wish, desire, want” and strengthened ᴹ√MBER, unglossed but with derivatives like ᴹQ. meren(de)/N. bereth “feast, festival” and ᴹQ. merya/N. beren “festive, gay, joyous” (Ety/MBER, MER). It is not clear whether Tolkien intended these roots to be related; the entry for ᴹ√MBER first gave the root as ᴹ√MER with N. mereth and meren, only to change it to ᴹ√MBER, N. bereth and beren when ᴹ√MER “wish, desire, want” was introduced (Ety/MBER; EtyAC/MBER).
In both earlier and contemporaneous Silmarillion drafts of the 1930s, however, Tolkien used N. Mereth Aderthad for “Feast of Reuniting” (SM/329; LR/126, 253) and continued to use S. Mereth Aderthad in the Silmarillion narratives of the 1950s and 60s (S/113; WJ/34). Possibly also related was the revision of the name Beril “Rose” to S. Meril in the unpublished epilogue to The Lord of the Ring written from the end of the 1940s (SD/117; SD/126). Conversely, {√MED >>} √MER “wish for, want” appeared in the Quenya Verbal System of the 1940s (PE22/102; 103 note #23) and Q. mer- “hope” appeared in the so-called “Merin Sentence”: Q. merin sa haryalyë alassë nó vanyalyë Ambarello “I hope that you have happiness before you pass from the world”, of unknown date but certainly written after the 1st edition of The Lord of the Rings (MS).
Thus it seems Tolkien merged ᴹ√MBER back into ᴹ√MER, though the meaning of the combined root isn’t clear given its various derivatives with senses like “feast”, “rose” and “wish, hope”. The last of these probably remained the core meaning, perhaps with “festive” extrapolated from “hopeful”.