It depends on the situation mostly. For the momentum-position uncertainty $\Delta p \Delta x\ge\frac{\hbar}{2}$ it is only conserved for systems who are in an eigenstate, that is in one of the fundamental functions that you can use to decompose the wavefunction. One can deduce that under eigenstates the energy-time uncertainty is 'undefined' as $\Delta E=0$ and $\Delta t$ is diverging to infinity, because energy can't change, simply by showing only one energy can exist in an eigenstate.
In non-eigenstates, such as the case with the free Gaussian wave packet, $\Delta p\Delta x$ are not conserved as noted by others, it linearly grows. However, several eigenstates exist, thus you have probability over energies so $\Delta E$ is now non zero. It does however take some rigorous math.
Because the energy in that example is directly proportional to the square of the momentum, and that distribution does not change, one can show that $<E>,<E^2>,\Delta E$ also do not change in the case of gaussian wavepacket. This is exactly what we get in classical mechanics as well, if you do not put energy in the system, there is no reason that it should. If $\Delta t$ is the time it takes to change said energy, the energy-time uncertainty relation is meaningless, as $\Delta t$ would have to be very long, and up to this point I assumed potentials that do not change in time.
With time dependent potentials, like an electromagnetic wave, it may cause change in the energy distribution, and thus $\Delta E$, but in accordance to the time-energy uncertainty. $\Delta E$ Can't change too much too fast, there is a limit to it, and that is the energy-time uncertainty.
Some other relations like the angular momentum also have similar relations.
In summary, the relations:
$$\Delta p\Delta x\ge\frac{\hbar}{2}$$
$$\Delta E\Delta t\ge\frac{\hbar}{2}$$
only speak about the minimal quantity the uncertainty can assume. In a closed system one might expect that only $\Delta E\Delta t$ to not change, while other quantities may change over time.
For the more rigorous stuff, showing : https://www.colorado.edu/physics/phys2170/phys2170_fa06/downloads/Gaussian.pdf