Tefilas Tal (and Geshem) were composed by Rabbi Eleazar Ha-Kallir who lived around 570 – 640 CE.
How did the synagogue service mark the change from winter to summer and vice versa before he composed these prayers?
The recitation of Tefillat Tal and Geshem are not dependent on Rav Hakalir, of course. While it is true that the majority of the piyut that's there was composed by him, there are still many minhagim that don't say either of these paryers.
This article focuses on Geshem (rain), but I would assume that the same principles could be applied to Tal.
Some highlights from the article:
O.C. 114:2 says that individuals should not begin saying Morid Hageshem until the gabbai or shat"z specifically announces this in the congregation. It further clarifies that someone who is ill and prays alone, should, therefore, not recite it in is silent amidah because it has not yet been announced publicly in a congregation. (It seems that such a person would have to wait until Mincha.) Note that O.C. makes no mention of Tefilat Geshem as this is a supplemental piyut.
The article states that Geshem (and I assume Tal) is not mentioned in many ancient siddurim and is not in some more "popular" ones such as Avodat Yisrael, Hertz, Hirsch or Otzar Hatefillot.
Sefer Haminhagim (c. 1300) is one of the first sources that mentions reciting Geshem
Sefer Shibolei Haleket from about the same era says that the shat"z just announces Mashiv Haruach prior to Musaph in a fashion similar to the announcement of the day of Rosh Hodesh. Another custom he mentions is that the congregation doesn't say it in their silent amidah and in the repetition, the chazan just says it (I infer as part of the 2nd bracha Mechalkel Chaim) followed by Livracha Velo Likelala, etc.