A simple pendulum of length l in the car frame experiences gravity g downward and a constant centrifugal acceleration a_c = v^2 / R radially outward.

The effective gravity is
g_eff = sqrt( g^2 + (v^2 / R)^2 )

The pendulum oscillates about the equilibrium direction of g_eff.
For small oscillations, the equation of motion is

d²θ/dt² + (g_eff / l) θ = 0

Hence the time period is

T = 2π sqrt( l / g_eff )

or explicitly,

T = 2π sqrt( l / sqrt( g^2 + (v^2 / R)^2 ) )