I think the formula will be essentially the same.

The formula you wrote is valid in general for the exponential map of analytic manifolds equipped with an analytic affine connection. It is stated and proved in this paper by Helgason (see pages 6-7 of the linked .pdf): *Some remarks on the exponential mapping of an affine connection. Math. Scand. 9 (l961), l29-l46.*

The existence of such connections on reductive homogeneous spaces, with an additional invariance property (that you might probably need at some point) is stated and proved in *K.Nomizu, Invariant affine connections on homogeneous spaces, Amer. J. Math. 76 (1954), 33-65*.