No, that is not true: the property of being nef, ample, etc. is not stable under specialization (although it is stable under generization). For instance, begin with $Y=\mathbb{P}^2_{\mathbb{Z}}$, i.e., $\text{Proj} \ \mathbb{Z}[s,t,u]$ together with its natural projection $$\pi:Y\to \text{Spec}\ \mathbb{Z}.$$ Consider the homogeneous ideal $$I = \langle st(t-s),s(u-pt),t(u-ps),u(t-s),u(u-ps) \rangle.$$ The corresponding zero scheme is the union of three disjoint sections of $\pi$, namely $[1,0,0]$, $[0,1,0]$ and $[1,1,p]$. Let $\nu:X\to Y$ be the blowing up of $Y$ along $I$.
There is a canonically defined pullback map of invertible sheaves, $$f:\nu^*\omega_{Y/\mathbb{Z}} \to \omega_{X/\mathbb{Z}},$$ which identifies $\nu^*\omega_{Y/\mathbb{Z}}$ with $\omega_{X/\mathbb{Z}}(-\underline{E})$ for a unique Cartier divisor $\underline{E}$ on $X$, the exceptional divisor of $\nu$. Now consider the invertible sheaf $L = (\nu^*\mathcal{O}_Y(2))(-\underline{E})$.
Over $\mathbb{Z}[1/p]$, the invertible sheaf $L$ is nef, and even globally generated. Essentially this is because the associated ideal $I[1/p]$ is generated by the homogeneous generators of degree $2$; indeed, the one cubic generator $st(t-s)$ is already in the ideal generated by the quadratic generators, $$st(t-s) = \frac{1}{p}\left(s[t(u-ps)]-t[s(u-pt)]\right).$$ However, in characteristic $p$ this is not nef: the strict transform of the line $Z(u)$ has intersection number $-1$ with $L$.
Edit. Replace "openness" claim by "stable under generization", see comments below.