In Sindarin, the combination ðv became ðw. The clearest description of this sound change appears in notes on the development of stops + nasals from the late 1960s:

tm, dm > ðm, later ðv, ðw (VT42/26).

This note only shows the development for v arising from the spirantalization of m, but another late note hints at similar developments for v arising from primitive b:

In Sindarin zd > ´d, đ but zg, zb > đʒ, đƀ > đa, đu, as in nadha “fetter” [< ✶nazg-], maða “mud” [< ✶mazgō/ŭ], buðu “large fly” [< ✶buzbō]. (PE19/101; the primitive forms appear earlier on the same page with Quenya derivations.)

This note shows zb > đƀ [ðβ] > đu, but the example makes it clear that resulting u occurs where the spirant became final: ✶buzbō [> *buđƀ(o)] > buðu (Tolkien generally used ƀ to represent IPA [β]). A plausible series of sound changes would be -ðƀ(o) > -ðv > -ðw > -ðu: it is normal for final [w] to became [u] after a consonant. I think the most likely set of developments are zb, db > ðƀ > ðv, which blends with dm > ðv, all of which later becomes ðw. The resulting -ðw > -ðu when final.

Some rough etymological notes discussing Quenya and Sindarin words for “throne” from the late 1960s provide some more examples of this sound change. These notes describe many possible developments, but the ones relevant to this discussion are:

khadmā. χanmā > χanw̃ā > hanw̃a. [Q.] tarhanwa “throne”, high seat.
ara. ... [S.] arahaðm. arahaðw, archaf. chaðw. haðw.
... [S.] haðma. haðwa, to seat. haðwad. arahaðwad (PE22/148).

These notes are somewhat difficult to parse, but it seems that the primitive form of “seat” was khadmā. Its Quenya derivative was hanwa and its Sindarin derivative was:

• khadmā [> *chaðv(a)] > chaðw > haðw [probably pronounced haðu].

The verb form haðwa- “to seat” indicates this development occurred medially as well, but here the pronunciation probably remained [ðw].

Note that one of the above forms, archaf, shows an inconsistent set of phonetic developments, and elsewhere on the same page Tolkien wrote “dm > v. gm > u̯w. dagma, dau(v).” It seems Tolkien was considering alternate developments whereby dm, gm > v, uw, which explains the alternate form archaf. An unrelated note from the late 1960s appearing along with The Shibboleth of Fëanor indicates that this alternate development might be North Sindarin:

In the Northern dialect, however, in final position only, CE. tw > dw > ðw, thw > þw, nw became b, v, f, m (VT41/8).

Tolkien’s notes associated with the Noldorin use of the Feanorian Alphabet from the 1930s indicate he at least considered similar developments at this early conceptual stage of the language:

The exilic development [of zb, zg] (to ðw, ði̯) suggests that W S became during ON [ðb, ðg], but these letters continued to be used. zd, however, became d, with lengthening of preceding vowel, and in consequence in late ON 2 was often substituted for @ (PE22/26).

Tolkien rejected these phonetic developments in the 1930s, however, because he decided [z] vocalized all before voiced stops in Old Noldorin, so that zb produced u̯b (PE22/26, note #78). Likewise, in Old Noldorin all voiced stops became nasals before nasals (not just in homorganic combinations of stops and nasals as in Sindarin), so that dm > nm. Thus, the combinations that normally produced ðv > ðw in Sindarin had distinct developments in Noldorin.

As the previous note from the 1930s indicates, the Sindarin sound change of v > w after ð resembles the medial post-consonantal vocalizations of [ɣ], whereby [ɣ] became [i] between sonants and vowels: lg, rg, ðg > lʒ, rʒ, ðʒ > li, ri, ði, probably passing through [j] (“y”) in the process. Unlike [ɣ], it seems [v] did not change after liquids, since medial and final lv, rv are fairly common (final lv, rv are denoted “lf, rf” in Tolkien’s orthography of Sindarin). Thus, only ðv > ðw.

Conceptual Development: As noted above, the combinations normally leading to ðv > ðw in Sindarin did not occur in Noldorin due to other, earlier sound changes. In Early Noldorin and Gnomish most (if not all) examples of dhw seem to be the result of dh + w, so it’s unclear how these phonetic developments played out in the earliest conceptual stages of the language.