For any $n\times n$ matrices $X_1$ and $X_2$, 
\begin{equation}
\begin{aligned}
	e^{X_1+X_2}-e^{X_1}&=\sum_{k=0}^\infty\frac1{k!}\,[(X_1+X_2)^k-X_1^k] \\ 
	&=\sum_{k=0}^\infty\frac1{k!}\;
	\sum_{(j_1,\dots,j_k)\in\{1,2\}^k\setminus\{(1,\dots,1)\}}\;
	\prod_{i=1}^k X_{j_i}, 
\end{aligned}
\end{equation}
so that 
\begin{equation}
\begin{aligned}
	\|e^{X_1+X_2}-e^{X_1}\|
	&\le\sum_{k=0}^\infty\frac1{k!}\;
	\sum_{(j_1,\dots,j_k)\in\{1,2\}^k\setminus\{(1,\dots,1)\}}\;
	\prod_{i=1}^k \|X_{j_i}\| \\ 
	&=\sum_{k=0}^\infty\frac1{k!}\;
	\sum_{m=1}^k \binom km \|X_1\|^m \|X_2\|^{k-m} \\  
	&=\sum_{k=0}^\infty\frac1{k!}\;[(\|X_1\|+\|X_2\|)^k-\|X_1\|^k] \\ 
	&=e^{\|X_1\|+\|X_2\|}-e^{\|X_1\|} \\ 
	&\le \|X_2\| e^{\|X_1\|+\|X_2\|}.\quad\Box 
\end{aligned}
\end{equation}