Need to know the exact values for sense resistor, resistors in op-amp and power supplies
Several points:

If you can make the power source for the "mobile" floating, as you show it being a battery, there is no need for a diff amp.  Declare the low side of the current sense resistor to be the ground of your measuring circuit.  The other side is then a nicely ground-referenced voltage proportional to the supply current.

As someone else already said, a 741 is a crappy opamp for this application, in fact most applications.  If you follow point #1, then you will need something with common mode input range extending to ground, or use a bipolar supply.

The signal you are looking for will likely be tiny compared to the steady current draw.  You therefore will need very good signal to noise ratio.

The current draw when not picking up a tweet probably varies over short times as a function of lots of other unpredictable things going on in the unit.  It is quite possible that these variations are larger than whatever incremental current draw there might be to process a tweet, if there even is one.  Again, this is all saying that the signal to noise ratio will be small.  Quite possibly the signal you want, if it even exists, will be so buried in noise as to be undetectable.