We investigate the problem of covert quickest change detection in a continuous-time setting, where a Brownian motion experiences a drift change at an unknown time. Unlike classical formulations, we consider a covert adversary who adjusts the post-change drift $μ= μ(γ)$ as a function of the false alarm constraint parameter $γ$, with the goal of remaining undetected for as long as possible. Leveraging the exact expressions for the average detection delay (ADD) and average time to false alarm (AT2FA) known for the continuous-time CuSum procedure, we rigorously analyze how the asymptotic behavior of ADD evolves as $μ(γ) \to 0$ with increasing $γ$. Our results reveal that classical detection delay characterizations no longer hold in this regime. We derive sharp asymptotic expressions for the ADD under various convergence rates of $μ(γ)$, identify precise conditions for maintaining covertness, and characterize the total damage inflicted by the adversary. We show that the adversary achieves maximal damage when the drift scales as $μ(γ) = Θ(1/\sqrtγ)$, marking a fundamental trade-off between stealth and impact in continuous-time detection systems.