In this paper we develop a sequential convex programming (SCP) framework for free-final-time covariance steering of nonlinear stochastic differential equations (SDEs) subject to both additive and multiplicative diffusion. We cast the free-final-time objective through a time-normalization and introduce per-interval time-dilation variables that induce an adaptive discretization mesh, enabling the simultaneous optimization of the control policy and the temporal grid. A central difficulty is that, under multiplicative noise, accurate covariance propagation within SCP requires retaining the first-order diffusion linearization and its coupling with time dilation. We therefore derive the exact local linear stochastic model (preserving the multiplicative structure) and introduce a tractable discretization that maintains the associated diffusion terms, after which each SCP subproblem is solved via conic/semidefinite covariance-steering relaxations with terminal moment constraints and state/control chance constraints. Numerical experiments on a nonlinear double-integrator with drag and velocity-dependent diffusion validate free-final-time minimization through adaptive time allocation and improved covariance accuracy relative to frozen-diffusion linearizations.