There was an interesting court case regarding a yeshiva and a shul in Elad in which it ended up that it was up to the courts to set the official time for starting davening.

According to Behadrei, City Hall of Elad allocated a plot to a shul in April 2000. 17 years later, in December 2017, the same City Hall (though under different leadership) allocated the same property for a yeshiva, Yeshivas Ateres Shlomo under the ownership and leadership of Rav Shalom Ber Sorotzkin.

Shul leadership tried to get the second allocation canceled after initial attempts at coming to a peaceful agreement of how to share usage of the building failed, as representatives of the yeshiva never cooperated and did not even go to the set meetings.

After spending nearly a year in court fighting over this, they came to agreements regarding most issues on the table with both institutions continuing to use the building. For example, they agreed that on Shabbos the shul membership would get use of 70 seats and the yeshiva would get 100 seats. During yeshiva vacation periods, the shul can use as much of the resources as they see fit.

Other issues as well were relatively easy to find solutions for. Such as, the shul would daven mincha 20 minutes before sunset while the yeshiva would daven mincha gedola at 12:30pm (and I guess 1:15 during the summer months). The shul and yeshiva would daven kabalas shabbos and maariv together, as well as davening mincha on Shabbbos afternoon 40 minutes before sunset together. On Motzei Shabbos the shul would daven first at their regular time and the yeshiva would daven after them. Expenses would be shared with shul membership paying a set monthly fee to the yeshiva to cover expenses.

Aliyot to the torah would divided equally among the yeshiva and the shul membership, as well as dividing up leading the various services. They also agreed upon an arrangement for sharing the gabbaus.

The one thing they could not agree on that required a court decision was what time to start davening on Shabbos morning. The shul wanted to start later, saying they always start at 8:30 with minor adjustments depending on the time of year and the time for shema. The yeshiva says that is too late, as yeshivas always start at 7:15. They also claimed that the shul historically did not really start at 8:30 but at 8:15 and is only claiming 8:30 for the purpose of negotiations. The shul said they would agree to make it earlier up to 8am, but as everyone gets up early to daven during the week before going to work, on Shabbos they want to start later.

Regarding that one issue they could not come to any agreement, so the courts decided that davening on Shabbos morning in the shul/yeshiva would be at 7:45am.