If L is a regular language then the language replace(L,σ,τ) is also regular
Let $A$ be the alphabet and let $a$, $b$, $c$, ... be the letters
(this is easier to type).

Hint. 
Consider the (non-deterministic) transducer $A^* \to A^*$ defined by $1 \xrightarrow{c \mid c}1\xrightarrow{a \mid b} 2 \xrightarrow{c \mid c} 2$
(more precisely, there are transitions $1 \xrightarrow{c \mid c} 1$ and $2 \xrightarrow{c \mid c} 2$ for any letter $c$ of $A$, including $a$ and $b$, but there is a single transition form $1$ to $2$). For each transition, the left hand part of the label is the input letter, the vertical bar is a separator and the right hand part is the output letter.