Autocorrelation of Addition of Two Independent Signals Given a random signal $ Z \left( t \right) $ which is addition of two independent signals $ X \left( t \right) $ and $ Y \left( t \right) $ with constant parameters $ a $ and $ b $:
$$ Z (t) = aX(t) + bY (t) $$
If the auto correlation function of $ R_{XX} \left( \tau \right) $ and $ R_{YY} \left( \tau \right) $ is known (Assume both are Stationary Signals).
What would be the Auto correlation function of $ Z \left( t \right) $?
Is it given by (I skipped the steps):
$$ {R}_{ZZ} \left( \tau \right) = a^2 R_{XX} \left( \tau \right) + b^2 R_{YY}\left( \tau \right) $$
I neglected the $ {R}_{XY} \left( \tau \right) $ contribution because it is zero as $ X \left( t \right) \perp Y \left( t \right) $ (Independent Random Signals).
You're correct as the Cross Correlation function vanishes.
This has the implicit assumption the process has zero mean (Actually, at least one of them).

Namely, in order to have $ {R}_{XY} \left( \tau \right) = 0 $ having $ X \left( t \right) \perp Y \left( t \right) $ isn't enough but at least of them has zero mean (Namely, $ \mathbb{E} \left[ X \left( t \right) \right] = 0 $, $ \mathbb{E} \left[ Y \left( t \right) \right] = 0 $ or both).

Pay attention that in practice for a single realization the empirical cross correlation won't vanish.