Is there anything in the Torah or the Talmud forbidding Judaism from establishing a call to prayer (of the type Islam has)?
There were various methods used to wake people for prayer and to announce the times.
For example, Prayer – waking people up to pray gives the following examples.
What happened when there was a need to wake up an entire tzibbur on a daily basis? During Medieval times, this was the of the shulklappers (a term first appears in a non-Jewish published in 1225) whose was to klap (knock) on peoples’ shutters and doors to indicate it was time to get up and shul. This was generally the shamash of the shul. The profession must have been common as surnames like Shulklapper and Klapper are still extant.
The custom of rousing people up for avodah and prayer extends all the way back to the Bais Hamikdosh. Among the people appointed over various duties in the Bais Hamikdosh, the Mishnah (Shekalim 5:1) counts Gevini the Announcer, whom, the Yerushalmi (ibid) explains, used to cry out every morning, “Rise, Kohanim, for the sacrificial service, Levi’im to sing, and Yisroel to stand by [as representatives of the nation].” ...
It has been claimed that the custom of waking people with a shulklapper died at the end of the 19th century and that its only remnant are antique shulklapper hammers on display in museums. This is not strictly true. The shulklapper persists in modern guise. Instead of old time shulklappers racing through the marketplace warning shops to close up, Shabbos niggunim now waft from loudspeakers to apprise people when Shabbos or Yom Tov are about to begin. Although funerals are no longer announced by a klap at the door, some communities still announce them from car-mounted loudspeakers.
And even Selichos have not been forgotten. Every year, R. Eizik Fried of Kiryas Belz makes the rounds of his Yerushalayim kehillah in his car. As a concession to modernity and perhaps due to the height of the apartment buildings, he has replaced the shulklapper’s hammer with a speaker on his car from which the ancient words still resonate: Yisroel am kodosh, kumu la.avodas haborei.
There is the recognition that different prayer times (within a range) may work for different people. The Talmud praises those who start the morning prayers super-early so they reach the middle part right at sunrise, however it's also acknowledged that's not something we can expect everyone to do. As long as you get those prayers in sometime in the three hours after sunrise, that's still acceptable. So there's no one time at which you would be shouting out to the town, "wake up for prayers!"
Similarly, the afternoon prayers could be anywhere from ~1pm until sunset. So you could sound an alarm "we've now reached the earliest time for afternoon prayers!", or "it's 15 minutes until the last possible time for prayers, hurry up if you haven't yet done so!", but there's no one set time.
(In something structured like a school dormitory however, everyone is expected to be at the structured prayer service, and there will be a wake-up call accordingly.)
There are towns in Israel that have sirens or the like to announce the Sabbath is about to start, on Friday afternoon. And the Talmud speaks similarly of trumpets to warn people that Passover is beginning soon, they have to get rid of their bread. But those are clearly-demarcated deadlines, whereas prayer times are ranges.