Concept:
$\text{BF}_3$ is a Lewis acid and $\text{NH}_3$ is a Lewis base.
They form a coordinate bond:
\[
\text{F}_3\text{B}\leftarrow\text{NH}_3
\]
Step 1: Hybridization around boron
After accepting the lone pair from nitrogen, boron forms four sigma bonds.
Therefore,
\[
\text{Steric number}=4
\]
\[
\Rightarrow sp^3 \text{ hybridization}
\]
with tetrahedral geometry.
Step 2: Hybridization around nitrogen
Nitrogen now also has four sigma bonds.
Hence,
\[
\text{Steric number}=4
\]
\[
\Rightarrow sp^3 \text{ hybridization}
\]
with tetrahedral geometry.
Therefore,
\[
\boxed{\text{B: }sp^3,\ \text{tetrahedral;\quad N: }sp^3,\ \text{tetrahedral}}
\]