$\newcommand{\mat}[1]{\begin{bmatrix}#1\end{bmatrix}}$ $\newcommand{\norm}[1]{\left\lVert#1\right\rVert}$Let $A$ be an $n\times n$ real matrix which has real block diagonal form and each block has a modified real Jordan normal form, i.e. $A = \operatorname{diag} \{J_i\}$ where $$ J_i = \mat{\lambda_i & d_i & & \\ & \ddots & \ddots & \\ & & \lambda_i & d_i \\ & & & \lambda_i} \tag{1}\label{1} $$ if $\lambda_i$ is real and $$ J_i = \mat{C_i & d_i I & & \\ & \ddots & \ddots & \\ & & C_i & d_i I \\ & & & C_i} \tag{2}\label{2} $$ if $\lambda_i = a_i \mp i b_i$ are complex conjugate pairs where $d_i > 0$ and $$ C_i = \mat{a_i & b_i \\ -b_i & a_i} $$ Let $\phi_A(t)=\int_0^t e^{A \tau} d\tau$ for a real square matrix $A$.
For a scalar smooth function $f(t)$, we have (around $0$) $$f(t) = f(0) + t f'(0) + \frac{t^2}{2!} f''(c)$$ for some $c \in (0,t)$, so-called Lagrange remainder. Unfortunately, it doesn't immediately generalize to matrices. However, we can still use the properties of $\phi_A(t)$ to use it for each element of the matrix. Since $\phi_A(0)=0$, $\phi_A'(0)=I$ and $\phi_A''(t)=e^{At}A$, for each Jordan block we can write $$\left(\phi_J(t) \right)_{ij} = t\delta_{ij} + \frac{t^2}{2!}\left( e^{Jc_{ij}}J \right)_{ij}$$ for some $c_{ij} \in (0,t)$ where $\delta_{ij}$ is the Kronecker delta. Now we can consider several cases for the induced 2-norm of the remainder $R_J := \left( e^{Jc_{ij}}J \right)_{ij}$.
Case I. $J = \lambda$ is real scalar. Then $\norm{R_J} \leq |\lambda| e^{\max(0,\lambda)t}$.
Case II. $J = C$ and $\lambda = a \mp ib$. In this case one can show that $\left|\left( e^{Jc_{ij}}J \right)_{ij}\right| \leq |\lambda| e^{\max(0,a)t}$. At this point we can use the maximum norm. Since $\norm{M} \leq n \norm{M}_\max$ for any square matrix where $\norm{M}_\max := \max_{ij} |M_{ij}|$, we have $$\norm{R_J} \leq 2 |\lambda| e^{\max(0,\operatorname{Re}(\lambda))t}$$
Case III. Let $J$ be as in $\eqref{1}$. In this case one can show that $\left|\left( e^{Jc_{ij}}J \right)_{ij}\right| \leq e^{\max(0,\lambda)t} f_{ij}$ where $$f_{ij} := \begin{cases} \frac{(dt)^{j-i-1}}{(j-i-1)!} \left( d + |\lambda| \frac{dt}{(j-i)} \right) & j > i \\ |\lambda| & j = i \\ 0 & j < i \end{cases}$$ If we bound $t$ such that $1 \geq dt$, then $f_{ij} \leq |\lambda| + d$. Remembering the bound on $t$, now we have $$\norm{R_J} \leq \dim(J) \left(|\lambda| + d\right) e^{\max(0,\lambda)t} $$
Case IV. Let $J$ be as in $\eqref{2}$. Similarly, bounding $t$ such that $1 \geq dt$ we have $$\norm{R_J} \leq \dim(J) \left(|\lambda| + d\right) e^{\max(0,\operatorname{Re}(\lambda))t} $$
So, overall we have $$ \norm{R_A} \leq e^{\eta_0(A)t} \xi(A) $$ if $t < \tau$ where $\eta_0(A) := \max_i\{0, \operatorname{Re}(\lambda_i)\}$, $\xi(A) := \max_i \{\dim(J_i) \left(|\lambda_i| + d_i\right) \}$ and $\tau := \min_i \{d_i^{-1}\}$.
Is there any problem with this approach? Did I miss anything? Can we do better in terms of conservatism or the bounds on $t$?
