Different conductors in the electical circuit
An extreme simplification of your situation is represented in the following circuit:







simulate this circuit &ndash; Schematic created using CircuitLab

where \$V_s\$ represents the voltage delivered by the power company, \$R_w\$ the wiring resistance and \$R_{load}\$ your home appliances.

If the energy meter is placed before both \$R_w\$ and \$R_{load}\$, it will measure the power consumed by both resistor:

$$
P = \frac{V_s^2}{R_w+R_{load}}
$$

If you lower \$R_w\$ the power \$P\$ will be higher and so will the recorded energy consumption by your household (but see below).

What will change in your favor is the efficiency of your electricity bill: more of the consumed power will go into powering your appliances.

In other words, there will be less drop on \$R_w\$ hence \$R_{load}\$ will experience an higher voltage across it. This means that, for example, traditional light bulbs will shine a little brighter, giving out more lumens, i.e. more light. Other appliances will give a little better performance too: an electric heater will warm up to the set temperature in less time, so it will turn off earlier (thus compensating in part your higher power consumption from the grid).

Of course the load of the electrical system is not necessarily purely resistive. Electronic loads with no PFC (power factor correction) will behave differently than a simple resistor. Anyway, if they get a slight more voltage across them, chances are that the current they will draw will be lower (sort of constant-power load), thus again compensating for your "theoretical" higher power draw from the grid.

EDIT (to integrate a relevant comment by Neil_UK)

As Neil has said in its comment, it is unlikely that the difference in your bill will be much different due solely by the different resistivity of the wires.

I'll add that most probably, the best rationale behind converting aluminium wiring to copper wiring is reliability. The wiring system will heat-up less, hence its insulation will be less stressed. Moreover copper is more flexible than aluminium and will endure mechanical stress (e.g., from thermal expansion) much more nicely. Even from a corrosion standpoint copper is better, because its oxide is mildly conductive whereas aluminium oxide is quite a good insulator.