Switch to battery when voltage supply drops certain threshold
You can obtain a sharp (or at least sharper) transition between \$V_{EXT}\$ and \$V_{BAT}\$ respect to what happens in the current version of circuit by modifying it in the following way:







simulate this circuit &ndash; Schematic created using CircuitLab

In this circuit, 


when \$V_{EXT}\$ lies below the threshold voltage level \$V_{th}\simeq V_Z+V_{BEQ_{1on}}\$, \$Q_1\$ is off and \$Q_2\$ is saturated: this implies that \$V_{GSM_1}\ll V_{GS_{on}}\$ \$\implies M_1\$ is on and thus the current flows from the battery to the load.
when \$V_{EXT}\$ rises above the threshold voltage \$V_{th}\$, \$Q_1\$ quickly goes on and \$Q_2\$ goes progressively out of saturation. The large gain of this two stage circuit forces a quick rise of \$V_{GSM_1}\$ even for relatively small values of \$V_{EXT}-V_{th}\$: so \$Q_2\$ goes quickly off, thus forcing \$M_1\$ off, since in this condition \$V_{GSM_1}\simeq 0&gt; V_{GS_{on}}\$.


Notes


The large gain of the two-stage transistor amplifier is responsible for the sharp switching between the two power sources: however, note the that the switching speed of \$M_1\$ is controlled by its gate charge. If switching times of the order of nearly several hundred of microseconds are too much for your application, you should optimize the output circuit of \$Q_2\$ in order to reach the sought for switching speed.
If needed, you could change the \$Q_1-Q_2\$ amplifier layout to realize a Schmitt trigger: this is accomplished by connecting together the emitters of \$Q_1\$ and \$Q_2\$ and inserting a resistor going from their common node to reference ground. This may be useful if you need some hysteresis in the switching levels for \$V_{EXT}\$