A root for “laugh” appearing in notes written around 1959 (PE17/159), likely connected to S. Lalaith “Laughter”, the name of Túrin’s deceased sister (S/198). Some possible precursors appeared in the Qenya Lexicon of the 1910s: unglossed ᴱ√LALA whose derivatives had to do with babbling (QL/50), and ᴱ√KAKA “laugh” with derivatives in both Qenya and Gnomish (QL/44; GL/24).
Tolkien gave a different primitive form ✶glada- as the basis for “laugh” words in The Shibboleth of Fëanor from the late 1960s; this produced S. glað- but still resulted in Q. lala- (PM/359). This new primitive had two problems (1) it is not compatible with S. Lalaith and (2) it shows a rather unusual assimilation of primitive d to preceding l in Quenya rather than the normal change to r: ✶(g)lada- > laða- > Q. lala- instead of ✱✱lara-. Elsewhere such assimilation did not occur, for example Q. lerembas < ✶led(e)mbasse (PE17/52).
Thus for purposes of Neo-Eldarin I think it is preferable to assume √LAL was the root for “laugh” words. However, I also think it is worth retaining √GLAD as a Sindarin-only root for similar concepts, perhaps “guffaw” and “joke”, given the limited semantic space we have in Elvish roots.
lala-(1) vb. "laugh" (PM:359), possibly with pa.t. *landë because the stem is given as g-lada-.