$\newcommand{\vpi}{\varphi}\newcommand\Z{\mathbb Z}$For real $t\ge0$, let $X_t:=mt+W_t$, where $m$ is a real number and $W_\cdot$ is a standard Brownian motion. So, $X_\cdot$ is a drifted Brownian motion starting at $0$ with the constant drift coefficient $m$. 

For real $c$, let $T_c:=\min\{t\ge0\colon X_t=c\}$. The probability in question is 
\begin{equation*}
	P_{t,a,b,m}:=P(T_a\le t,T_b\le t)=1+P(T_a>t,T_b>t)-P(T_a>t)-P(T_b>t), 
\end{equation*}
where $-\infty<a<0<b<\infty$ and $t>0$. 
By rescaling, without loss of generality $t=1$, because $P_{t,a,b,m}=P_{1,\,a/\sqrt t,\,b\sqrt t,\,m\sqrt t}$. So, it is enough to find 
\begin{equation*}
	P_{a,b,m}:=P_{1,a,b,m}=1+Q_{a,b}-Q_{a,\infty-}-Q_{-\infty+,b}, \tag{1}
\end{equation*}
where 
\begin{equation*}
\begin{aligned}
	&Q_{a,b} \\
	&:=P(T_a>1,T_b>1) \\
	&=P(a<X_s<b\ \forall s\in[0,1]) \\
	&=P(a-ms<W_s<b-ms\ \forall s\in[0,1]) \\ 
	&=\int\limits_{a-m}^{b-m}P(a-ms<W_s<b-ms\ \forall s\in[0,1],W_1\in[x,x+dx])) \\ 
	&=\int\limits_{a-m}^{b-m}P(a-(m+x)s<W_s-sW_1<b-(m+x)s\ \forall s\in[0,1], \\
	&\, \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad W_1\in[x,x+dx])) \\ 
	&=\int\limits_{a-m}^{b-m}P(a-(m+x)s<W_s-sW_1<b-(m+x)s\ \forall s\in[0,1]) \\
	&\, \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad P(W_1\in[x,x+dx]) \\ 
	&=\int\limits_{a-m}^{b-m}P(a-(m+x)s<W_s-sW_1<b-(m+x)s\ \forall s\in[0,1], \\
	&\, \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad W_1\in[m+x,m+x+dx]) \\ 
	&\, \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\frac{P(W_1\in[x,x+dx])}{P(W_1\in[m+x,m+x+dx])} \\ 
	&=\int\limits_{a-m}^{b-m}P(a<W_s<b\ \forall s\in[0,1], \  W_1\in[m+x,m+x+dx])  
\,\frac{\vpi(x)}{\vpi(m+x)}, 
\end{aligned}
\end{equation*}
where $\vpi$ is the standard normal pdf; the third and second equalities from the end of the above multiline display follow by the independence of the Brownian bridge $(W_s-sW_1)_{s\in[0,1]}$ from $W_1$. 

By multiple reflection (Lévy's triple law -- see e.g. [Proposition 6.10.6][1] or [Theorem 6.18][2]), 
\begin{equation*}
	P(a<W_s<b\ \forall s\in[0,1], \  W_1\in[m+x,m+x+dx])=q_{a,b}(m+x)\,dx,
\end{equation*}
where 
\begin{equation*}
	q_{a,b}(z):=\sum_{k\in\Z}[\vpi(z+2kh)-\vpi(2b-z+2kh)],
\end{equation*}
\begin{equation*}
	h:=b-a,
\end{equation*}
and $z\in(a,b)$. 
Taking now the the latter integral in the above multiline display, we get 
\begin{equation*}
\begin{aligned}
	Q_{a,b}&=Q_{a,b;m}:=\sum_{k\in\Z}e^{-2 h k m} (\Phi (a+2 h k+h-m)-\Phi (a+2 h k-m)) \\ 
&	-\sum_{k\in\Z}  e^{2 m (a+h k+h)} (\Phi (a+2 h (k+1)+m)-\Phi (a+2 h k+h+m)),  
\end{aligned}
\tag{2}
\end{equation*}
where $\Phi$ is the standard normal cdf. 

Now one can find the limits $Q_{a,\infty-}$ and $Q_{-\infty+,b}$ of $Q_{a,b}$ as $b\to\infty$ and as $a\to-\infty$, respectively. Alternatively, these limits can be found directly -- similarly to (but more simply than) $Q_{a,b}$. 

Thus, by (1), we have obtained an explicit expression for $P_{a,b,m}$. 

---

Here is the graph $\{(m,Q_{a,b;m})\colon |m|<5\}$ for $a=-1$ and $b=2$: 

[![enter image description here][3]][3]

The two series in (2) converge very fast: for $a=-1$ and $b=2$, the maximum absolute error seems to be $<2\times10^{-12}$ if the two summations $\sum_{k\in\Z}$ there are each replaced by $\sum_{k=-1}^1$. 


  [1]: http://www.fcfm.buap.mx/jzacarias/cursos/procesos/libros/book3pe.pdf
  [2]: https://books.google.com/books?id=Ph7oBQAAQBAJ&pg=PA71&lpg=PA71&dq=%22L%C3%A9vy%27s+triple+law%22&source=bl&ots=1raHTwJyC3&sig=ACfU3U1Whb9t_f4d85hdGS1-VKL4U4e08w&hl=en&sa=X&ved=2ahUKEwjg-rLknpj1AhXmFTQIHVuFDuMQ6AF6BAgZEAM#v=onepage&q=%22L%C3%A9vy's%20triple%20law%22&f=false
  [3]: https://i.sstatic.net/pktdd.png