Question about time constant of RC circuit?
Yes, it will not change to a significant value. Here is the answer for why it is so!

If you calculate the the RC time constant for this circuit it will be 

Tr=(R1||RV1).C1

Since in the parallel combination of two resistors, the effective value will be very close to the smaller resistance value among the two,hence it dominates. So here in your circuit the resistor R1(10k) is dominating and the slight change in the value of RV1 will not affect the effective parallel value to a great extent and hence your time constant remains almost the same.