Firstly Q1 is an emitter follower, not a switch (you need a PNP device in common-emitter configuration for a switch). The problem with what you have is the LM358 won't be getting 5V, possibly about 3.5 to 4V only.
Secondly the device driving Q1 cannot output anything like 5V in the first place, the LM358 is not rail-to-rail and will probably only raise its output to 3.5V or so. So the second opamp will not get enough voltage to function.
Thirdly there is no decoupling on the second opamp - some opamps require this.
The opamp gain stage current has only x2 gain, which presumably is not the intention, more like x10 to x100 is usually needed for a 5mΩ shunt.
And lastly the shunt amplifier should be a differential amplifier circuit connected to both sides of the shunt (in a 4-terminal or "Kelvin" connection). Otherwise the stray resistance of the wiring in the high current circuit will throw out you measurement significantly.
Why are you using one LM358 to switch on and off another? You still have to power at least one opamp all the time, you haven't gained anything!
The first device should be a comparator anyway, not an opamp, comparators compare voltages, opamps amplify.
I suggest first off forget the flawed power-saving stuff and concentrate on getting the differential amplifier working. The if you need to reduce the power consumption have a look at the "high side transistor switch" section here: https://www.baldengineer.com/low-side-vs-high-side-transistor-switch.html