When $N>1$ you cannot control the oscillation by the total variation, because it would mean that $BV$ functions are locally bounded. However, a $BV$ function can have essential supremum equal $+\infty$ and essential infimum equal $-\infty$ on every open set. In the case of vector field this phenomenon may apply to each component of the vector field. You can find an example here https://mathoverflow.net/a/321502/121665. This example is for Sobolev functions, but Sobolev functions are in $BV$.