How to quantify delay in filtered output?
It's the transient and the group delay which are associated with the initial conditions and the LTI filter phase response $\phi(w)$ respectively.

The group delay (in samples) associated with the LTI filter is:
$$\tau = - \frac{d \phi(w) }{dw} $$

The output will be shifted by $\tau$ samples wrt to the input. You shall compansate this shift for synchronization.

This delay is constant for linear phase filters which are of symmetric FIR type filters. For IIR type filters, since their phase response is not linear therefore getting a fixed group delay is not possible.