Did the writers of vedanta and purva mimansa sutras, badrayana and jaimini write about the eternal nature of the Vedas?
Yes , Vedanta Sutras or Brahma Sutras is stating that Vedas are eternal. It's mentioned in Adhyaya 1 -Pada 3-Sutra 29 & 30 - So Badrayana the author of Brahma Sutras agrees that Vedas are eternal.
अत एव च नित्यत्वम् ॥ 1.3.29 ॥
ata eva ca nityatvam || 29 ||
From this very reason also (results) the eternity (of the Vedas). Since the objects are eternal, that is, gods etc. as types are eternal, the Vedic words are eternal. This establishes the eternal nature of the Vedas. The Vedas were not written by anybody. They are impersonal and eternal. The Rishis only discovered them but were not authors of the Vedic texts.
समाननामरूपत्वाच्चावृत्तावप्यविरोधो दर्शनात् स्मृतेश्च ॥ ३० ॥
samānanāmarūpatvāccāvṛttāvapyavirodho darśanāt smṛteśca || 30 ||
And because of the sameness of names and forms (in every fresh cycle) there is no contradiction (to the eternity of the Vedic words) even in the revolving of the world cycles, as is seen from the Sruti and the Smriti.
An objection is raised. Since at the end of a cycle everything is completely destroyed and creation begins afresh at the beginning of the next cycle, there is a break in the continuity of existence; so even as types the gods are not eternal. This upsets the eternal relation of Vedic words and the objects they represent, and consequently the eternity of the Vedas and their authority fall to the ground. This Sutra refutes it. Just as a person after waking from deep sleep finds no break in the continuity of existence, so also in the state of Pralaya (end of a cycle) the world is in a potential state—in seed form—in ignorance, and not completely destroyed; at the beginning of the next cycle it is again manifested into a gross form with all the previous variety of names and forms. As the world does nqt become absolutely non-existent, the eternity of the relation between Vedic words and their objects is not contradicted, and consequently the authoritativeness of the Vedas remains. This eternal existence of the world in gross and fine forms alternatively and the similarity of the names and forms are brought out by the Sruti and Smriti texts. “As formerly the Lord ordered the sun and the moon, heaven,, earth, the sky” etc. (Rig-Veda 10 . 190. 3).
Purva Mimamsa Sutra of jaimini
The Purva Mimamsa school accepts Shabda (sound) as valid source of knowledge (Pramana) . jaimini in his Purva Mimamsa sutras state that The Veda is an instance of Sabda Pramana. The eternality of vedas can be proved by eternality of words as it believes that Words are eternal. The Mimamsa school asserts that since the Vedas are composed of words (shabda) and the words are composed of phonemes, the phonemes being eternal, the Vedas are also eternal.
In order to answer Another argument in support of view that the Veda is not eternal and Veda has had a beginning in time jaimini after proving eternity of vedas gives this answer. He is mentioning the eternity of Shabda and consequently of Vedas in this sutra. Adhyaya 1 Pada 8 - Sutra 29 of jaimini's Purva Mimamsa Sutras.
उकन्तु शब्दपुर्वत्वम् |1.8.29|
But the eternity of word has already been established.
In answer to above argument it is declared that arguments in support of the eternity of all words (and the vedas ) have already been advanced above under sutra 6-23 , where it has been proved that all word are eternal . All that is necessary therefore , on the present ocession is to answer the arguments put forward by the opponent in sutra 27 and 28.
All such and other arguments are refuted by jaimini to prove the Apurushya and eternal nature of vedas .
And here in this sutra he is indirectly mentioning about Eternity of Vedas.