Non-Hermitian systems driven along slow parametric loops undergo non-adiabatic transitions whose outcome depends sensitively on the driving speed, yet no explicit formula has been available for the critical timescale T_cr at which these transitions develop. Using a 2× 2 Hamiltonian with circular parameter trajectories, we derive T_cr = G ln(1/|Δ|) in closed form for non-encircling loops, phase-shifted loops, offset loops, and loops encircling exceptional points, where G is a geometry-dependent growth factor and Δ is the instability seed. This formula sharply separates the regime where the system remains in the averagely dominant eigenstate (T< T_cr) from the superadiabatic regime where the instantaneous dominant eigenstate takes over (T> T_cr), resolving the apparent tension between the previous literature. We identify two competing seeds: a geometric Stokes multiplier and the finite-precision floor. When the geometric seed vanishes, precision alone governs the transition, yielding T_cr ∝ mlnβ, linear in the number of precision bits m. This provides a purely forward-evolution manifestation of precision-induced irreversibility (PIR) , demonstrating that the fundamental limit identified through echo protocols also controls the outcome of slow non-Hermitian dynamics without requiring time reversal. For PT-symmetric energy spectra, T_cr additionally determines the onset of chirality: the dynamics is non-chiral for T< T_cr and chiral for T> T_cr.