Nusach for Tefillaus Tal v'Geshem

I don't think we have a recording of it. The only recording I know of is of Benno Weis.
We say the first two Borchos in the regular Shabbos melody, except for Mechalkel Chayim which is done in the melody of Tefillas Tal and Geshem (Oyom Zchor No etc.).

Michael

According to Minhag FFAM, the Kaddish before Musaf of Tefillas Tal and Geshem is in the Yomim Noro'im melody (Musaf of RH and YOK) until Le'eilo. From Le'eilo we change to the Tal and Geshem melody, and the Chazzan states out loud in that tune also "Mashiv Horu'ach Omorid Hagoshem", (and on Pesach "Morid Hatal" or "mechalkel Chayim Bechesed").
Ne'iloh tune is not used at all.
The Divrei Kehillos page 337 asks why don't we sing Atto Gibbor in the melody used on every Shabbos or Yom Tov which has Krovos. He answers that he assumes the reason we use the regular shabbos melody and not the Piyyut melody is so the Chazzan gets used to the change, and this should be done in the same tune he will be singing the Amidah in the coming Shabbossos.