Let us assume {R}_{1},{R}_{2},{R}_{3}&{R}_{4} is connected balanced wheat stone bridge, such that R3R1=R4R2...(1) as shown below,

If galvanometer and cell is exchanged, then circuit becomes as shown in below figure,

The circuit is redrawn and can be visualised as below figure,

In this case, if it is balanced, then there must be,
R2R1=R4R3...(2)
Equation (1)&(2) realises that, both equations are same, which means null point is not affected.