A root for “centre, middle”, it first appeared in The Etymologies of the 1930s as ᴹ√NED with augmented form ᴹ√ENED (Ety/ÉNED, NÉD). The only derivative of its unaugmented form was the Noldorin prefix N. nedh- “mid-”; its usual forms N. enedh/ᴹQ. ende were derived from the augmented form ᴹ√ENED. In notes from the 1950s and 60s, it seems this augmented form became the normal form, as in √EN(ED) “center, middle” (PE17/26) or ened- “middle” (VT47/26).
One complication in the later iterations of the root was the appearance of the word S. Enedwaith “Middle-region” in The Lord of the Rings maps and appendices (LotR/1089). I suspect this form dates from the period in Lord of the Rings drafts of the 1940s where Tolkien decided to represent [ð] as d: compare N. Fanuidol (TI/306) vs. S. Fanuidhol (LotR/283) and N. Galadrim (TI/239) vs. S. Galadhrim (LotR/341). For whatever reason, Enedwaith was never corrected, and in some places Tolkien said this was a mistake and the proper form should be Enedhwaith (PM/328; VT42/20) while in other places he considered revising the root for “middle” to √ENET or √HEN(ET) (VT41/16).
In his notes on numbers and fingers from 1968, Tolkien said the primitive form was ened with variant form enel (VT47/16, 29; VT48/9-10, 24-25), as the result of frequent d/l interchange in Common Eldarin (VT47/29). In these notes this “middle” root was likewise connected to √ENEK “three” as represented by the “middle” finger (VT48/10). In this set of 1968 notes Tolkien also said √ENET was connected to the distinct root √EN “again” and had the sense “one more” (VT47/15; VT48/25). As for the enel “middle” variant, it seems it was used more in Nandorin (VT47/29), but was a factor in some Quenya words such as Q. enelmo “go-between, intervener, intermediary, mediator” (VT47/14).
Neo-Eldarin: For purposes of Neo-Eldarin, I think it is best to assume the root for “middle” was mainly √ENED, and that Tolkien’s flirtations with √(H)ENET “middle” was a transient idea. I think it is also worth retaining the unaugmented prefix nedh- “mid-” for Neo-Sindarin words, though originating from a lost original vowel rather than a lack of augmentation.