EDIT: this is a new proof, in arbitrary characteristic, which uses comments and results of OP.
Consider the formal power series $$\exp(x)=1 + x+\frac{1}{2!}x^2+\dots$$ over ${\mathbb{Q}}$ and the polynomial $$\exp_{<n}(x)=1+x+\dots+\frac{1}{(n-1)!}x^{n-1},$$ which is defined also over ${\mathbb F}_p$ for $p\ge n$.
Theorem 1. Let $k$ be a field of characteristic 0 or of positive characteristic $p$. Let $H\subset G={\rm GL}_{n,k}$ be an algebraic subgroup defined over $k$. When ${\rm char}(k)=p>0$, we assume that $n\le p$. Let $X\in{\rm Lie}(H)\subset{\mathfrak{gl}}_{n,k} = M_n(k)$ be a nilpotent matrix. Then $\exp_{<n}(X)\in H(k)$.
Note that since $X\in M_n(k)$ is nilpotent, we have $X^n=0$.
Set $V=k^n$. Write $T^{r,s}(V)=V\otimes\dots\otimes V\otimes V^*\otimes\dots\otimes V^*$ ($r$ times $V$ and $s$ times $V^*$), where $V^*$ is the dual space to $V$. We set $$W=W^{\le N}=\bigoplus_{r,s=0}^N T^{r,s}(V).$$ Let $\theta=\theta^{\le N}$ denote the natural representation of $G={\rm GL}(V)$ in $W$, and let $d\theta$ denote the corresponding representation of ${\rm Lie}(G)={\mathfrak{gl}}(V)$ in $W$. Since $X$ is nilpotent and $\theta\colon {\rm GL}(V)\to {\rm GL}(W)$ is a homomorphism of linear algebraic groups, the linear operator $(d\theta)(X)\in{\rm End}(W)$ is nilpotent; see Springer, Linear Algebraic Groups (2nd ed.), Theorem 4.4.20. Note that in general in characteristic 0 it is not true that $((d\theta)(X))^n=0$, but one can show that $((d\theta)(X))^{2nN}=0$. In characteristic $p$, since the differential $d\theta$ of the homomorphism $\theta$ of algebraic groups is compatible with the $p$-operation (see Springer's book, Proposition 4.4.9), we have $((d\theta)(X))^p=(d\theta)(X^p)$. Since $n\le p$ and $X^n=0$, we have $X^p=0$, and hence, $((d\theta)(X))^p =0$.
Theorem 2. In characteristic 0, for any nilpotent matrix $X\in {\mathfrak{gl}}(V)= {\rm End}(V)$ we have $$\exp((d\theta)(X))=\theta(\exp(X)).$$
Note that $\theta(\exp(X))$ and $\exp((d\theta)(X))$ are defined because both $X\in {\rm End}(V)$ and $(d\theta)(X)\in{\rm End}(W)$ are nilpotent operators and because ${\rm char}(k)=0$.
Theorem 3. In positive characteristic $p$, for any nilpotent matrix $X\in {\mathfrak{gl}}(V)= {\rm End}(V)$, where ${\rm dim}(V)\le p$, we have $$\exp_{< p}((d\theta)(X))=\theta(\exp_{< p}(X)).$$
We deduce Theorem 1 from Theorems 2 and 3. There exists a natural number $N=N_H$ and a tensor ${{\mathfrak t}}={{\mathfrak t}}_H\in W=W^{\le N}$ such that $H$ is the stabilizer in $G$ of the line $k\cdot {{\mathfrak t}}\subset W$ with respect to $\theta$ and such that ${\rm Lie}(H)$ is the stabilizer in ${\rm Lie} (G)$ of this line with respect to $d\theta$; see Springer's book, Lemmas 5.5.1 and 5.5.2. Since $X\in {\rm Lie}(H)$, we have $(d\theta)(X)\cdot {{\mathfrak t}}=\lambda {{\mathfrak t}}$ for some $\lambda\in k$, and we have $\lambda=0$ since $(d\theta)(X)$ is nilpotent. Now in characteristic 0 it follows from Theorem 2 that $$\theta(\exp(X))\cdot {{\mathfrak t}}=\exp((d\theta)(X))\cdot {{\mathfrak t}}={{\mathfrak t}},$$ and therefore, $\exp(X)\in H(k)$, which proves Theorem 1 in characteristic 0. Similarly, in characteristic $p$ it follows from Theorem 3 that $$\theta(\exp_{< p}(X))\cdot {{\mathfrak t}}=\exp_{< p}((d\theta)(X))\cdot {{\mathfrak t}}={{\mathfrak t}},$$ and therefore, $\exp_{<n}(X)=\exp_{<p}(X)\in H(k)$, which proves Theorem 1 in characteristic $p$.
Proof of Theorem 2. Let $A={\rm GL}(V)$ , where $V=k^n$ and $k$ of characteristic 0. Let $X\in \mathfrak{gl}_{n,k}={\rm End}(V)$ be a nilpotent matrix. We define the exponential polynomial map $Z(t)=\exp(tX)\in A(k)$ by the polynomial $\exp_{< m}$ for some $m\ge n$, where the result does not depend on the choice of $m$. Then $Z(t)$ is the solution of the differential equation $\frac{d}{dt} Z(s)=X\cdot Z(t)$ with initial condition $Z(0)=1_A$. Now let $\theta\colon A\to B$ a homomorphism of algebraic $k$-groups, where $B={\rm GL}(W)$. Then the following diagram commutes: $$ \require{AMScd} \begin{CD} {\rm Lie}(A) @>{d\theta}>> {\rm Lie}(B);\\ @V{e_A}VV @VV{e_B}V \\ A @>{\theta}>> B,; \end{CD} $$ where the polynomial maps $e_A$ and $e_B$ are given by $\exp_{<M}$ for any $M>{\rm max}({\rm dim}(V),{\rm dim}(W))$. Indeed, both composite polynomial maps are solutions of the same differential equation with the same initial condition, and the assertion follows from the uniqueness theorem for polynomial solutions of first-order linear differential equations, see Mike Crumley's paper, Lemma 5.3(3). This completes the proof of Theorem 2.
Proof of Theorem 3. Let $A={\rm GL}(V)$ , where $V=k^n$ and $k$ of characteristic $p>n$. Let $X\in \mathfrak{gl}_{n,k}={\rm End}(V)$ be a nilpotent matrix. Then $X^n=0$ and therefore, $X^p=0$. We define the exponential map $Z(t)=\exp(tX)\in A(k)$ by the polynomial $\exp_{< p}(tX)$. The $Z(t)$ is the solution of the differential equation $\frac{d}{dt} Z(s)=X\cdot Z(t)$ with initial condition $Z(0)=1_A$. Now let $\theta\colon A\to B$ a homomorphism of algebraic $k$-groups, where $B={\rm GL}(W)$. Then the differential $d\theta\colon {\rm Lie}(A)\to {\rm Lie}(B)$ is compatible with the $p$-operation, that is, $((d\theta)(X))^p=(d\theta)(X^p)$; see Springer's book, Proposition 4.4.9. Since $X^p=0$, we see that $((d\theta)(X))^p=0$, and hence, $\exp_{<p}((d\theta)(X))$ is defined. Then the diagram above commutes, where the polynomial maps $e_A$ and $e_B$ are given by $\exp_{<p}$. Indeed, both composite polynomial maps of degree $<p$ are solutions of the same linear differential equation with the same initial condition, and the assertion follows from the uniqueness theorem, see the same Lemma 5.3(3) in the same Mike Crumley's paper. This completes the proof of Theorem 3.