This root was variously connected to life and warmth throughout Tolkien’s life. The earliest iteration of the root was ᴱ√LAWA from the Qenya Lexicon of the 1910s whose meaning was described as “much same as KOẎO [have life], but used of a vegetable”; it had derivatives like ᴱQ. laule “life, mode of life”, ᴱQ. laute “living thing, (esp.) vegetable”, and ᴱQ. lauke “vegetable, plant (species)” (QL/52). In the contemporaneous Gnomish Lexicon it had derivatives like G. lauth “a plant, herb” and G. laug “(of plants) alive, having sap, green, vigorous” (GL/53).
In The Etymologies of the 1930s it was glossed “warm” with derivatives ᴹQ. lauka and N. lhaug of the same meaning (Ety/LAW). In the Quenya Verbal System (QVS) from 1948 it was glossed “abound” with a Quenya verb ᴹQ. lauta- of the same meaning (PE22/103).
The root √LAW appeared in notes from the late 1950s serving as the basis for Q. loa “sun-year”, originally with the sense ✶lawā “growing, blooming”; it was also explicitly connected to its extended form √LAWAR with the sense “golden colour” (PE17/159). Its final appearance in currently published materials was in some Late Notes on Verbs from 1969, where it was given as √LAW “flourish (green), grow” with derivative Q. lauya- of the same meaning (PE22/156).
Despite all these minor variations, the general meaning of the root was fairly stable, having mainly to do with life and flourishing (especially of plants), and also connected to warmth and sunlight, probably by association with its extended form √(G)LAWAR “golden colour or light”, which applied to sunshine among other things.
A root meaning “grow, flourish” used in both Quenya and Sindarin, though in Quenya it was influenced by other roots such as √AL(A) “good, blessed”. Its precursors in the earliest versions of Tolkien’s languages seem to be ᴱ√ALA “spread” (QL/29) and a Gnomish-only root ᴱ√cala [kala], unglossed but with derivatives like G. calw “green shoot, sapling, sprout” and G. caltha- “wax, grow, flourish” (GL/25).
The root ᴹ√GALA first appeared in The Etymologies of the 1930s with various glosses, the last being “thrive” (Ety/GALA). The relevant entries went through considerable revision. For example, the meaning of this root was first given as “grow”, but this meaning was rejected and Tolkien said the verb for “grow” was ol- (EtyAC/GAL(AS)). This new verb seems to be based on a variant root ✱ᴹ√GOL, as indicated by ᴹQ. ola- “grow” < ᴹ✶golā- “grow” appearing in the Quenya Verbal System (QVS) from 1948 as a replacement for ᴹQ. ala- < ᴹ✶galā- (PE22/113, note #80), though the new root might also have been ᴹ√OL “grow” as indicated earlier in the same document (PE22/103). Since ᴹ√GALA was retained in The Etymologies, most likely the meaning of this root was changed from “grow” to “thrive” in the 1930s and 40s.
It seems ᴹ√OL “grow” survived into the 1950s (NM/84, 119-120), often with the sense “become” as well (PE22/134); see the entry on √OL for discussion. However, the sense “grow” as also restored to √GAL, but limited mainly to plants. For example, Common Eldarin: Verb Structure from the early 1950s had the primitive verbs ✶galā- “grow (of plants)” vs. ✶olā- “become, come into being, turn into (another state)” (PE22/134). This definition of √GAL as “growth” (along with “flourish” and “be healthy”) appeared regularly in Tolkien’s later writings, though it was often limited primarily to plants, especially in its connection to ✶galadā, the basis for tree words (PE17/25, 135, 153).