After you measured position $q$ at time $t$, even the probability distribution of the momentum you will measure at time $t+\Delta t$ will be "mostly unrelated" to the probability distribution of the momentum you would have measured at time $t-\Delta t$. And if you would measure position again at time $t+2\Delta t$, it will be "mostly unrelated" to both the actual measurement outcome and the probability distribution of the position measurement at time $t$.
You might still ask whether there is any limitation to how accurate you can measure momentum at time $t+\Delta t$, but if you measured position from time $t-\Delta T$ to time $t$ and momentum from time $t$ till time $t+\Delta t$, then this probably just boils down to how accurate you can make a momentum measurement in a finite time $\Delta t$. For a quantum object with mass $m$, this is probably given via the relation $E=\frac{p^2}{2m}$ by $\Delta E=\frac{p+\Delta p/2}{m}\Delta p$. Assuming $\Delta E\Delta t \geq\frac{\hbar}{2}$, then $\frac{p+\Delta p/2}{m}\Delta p\Delta t \geq\frac{\hbar}{2}$.