The two papers that explored first savings under uncertainty in a two-period setting are
Leland, H. E. (1968). Saving and uncertainty: The precautionary demand for saving. The Quarterly Journal of Economics, 82(3), 465-473.
Sandmo, A. (1970). The effect of uncertainty on saving decisions. The Review of Economic Studies, 37(3), 353-360.
They both deal with a general two period utility function $U(c_0,c_1)$ not necessarily additive. But both remark the following related to such a case:
The condition for precautionary savings depends on third cross partial derivatives, as well as on the third own derivative with respect to future consumption only.
If the utility function is additive $U=u(c_0)+Ev(c_1)$, then cross-partials are zero, so the result hinges only on the third own derivative for the second-period consumption.
It follows that with an additive over time utility function, it suffices that the second-period utility is quadratic (so third own derivative is zero), in order to not get precautionary savings, irrespective of the form of the first-period utility.