If the antecedents are mutually exclusive, then is the consequent true? Suppose I know that the following implications are true: $$P_1 \Longrightarrow (A \land B)$$ $$P_2 \Longrightarrow (A \land B)$$ for some premises $P_1, P_2$ and some conditions $A, B$. Does it follow that $A \land B$ is true, if $P_1$ and $P_2$ are mutually exclusive conditions?

No; given that $P_1$ and $P_2$ are mutually exclusive conditions, it is still possible that they're both false, and so we cannot deduce anything from $P_1\rightarrow Q,$ nor from $P_2 \rightarrow Q$. Explicitly (note that false implies false), the following is a counterexample to the conjecture:

$$P_1 = \mathrm{False}, \;P_2 = \mathrm{False}, \;Q = \mathrm{False}$$

Now on the other hand, if $P_1$ and $P_2$ are mutually _exhaustive_ conditions (i.e., at least one of them is true), and if we know that $P_1 \rightarrow Q$ and $P_2 \rightarrow Q,$ then we may deduce $Q$.