According to Einstein's theory of relativity, when two observers move relatively at different speeds, they will measure the time differently. This phenomenon is called time dilation. Specifically, when one observer moves at a speed close to the speed of light relative to another observer, the moving observer will feel that time passes more slowly.
But when astronauts and earthlings use mobile phones to time, the relative speed between them is much lower than the speed of light. So the time expansion effect caused by speed here is almost negligible.
Next, we consider the communication delay between the earth and the astronauts. The title refers to "live broadcast without delay", which means that the live broadcast signal is transmitted from the astronaut to the earth instantly without delay. Therefore, the communication delay will not affect the timing time displayed on their mobile phones.
In addition, the difference of gravitational field between the earth and astronauts may also affect the passage of time. However, considering that the Earth and astronauts are located near the Earth, the time difference caused by this gravity difference is extremely small, far less than the accuracy of mobile phone timing.
To sum up, because the relative speed of astronauts and earthlings is much lower than the speed of light, there is no communication delay, so when they use their mobile phones at the same time, the time displayed on the mobile phones should be the same. Of course, the premise here is that the timing function of the mobile phone itself is consistent and is not affected by other external factors.