@bitpop:  I don't understand why you think the electrician is an idiot.  Electrician <> Electrical Engineer.  It would not surprise me if there existed some industrial machinery that *did* remember its original voltage settings.  And why should the Electrician know how CPU power supplies are designed?  It sounds to me like he told his friend (the OP) "I don't want to be blamed if the higher voltage burns out your computers."

@snootch:  Nothing in this thread suggests the electrician is incompetent.  He should know what gauge to use for a 20A circuit.

@OP:  I will give your electrician the benefit of the doubt and assume he is doing everything by the book.  Most jurisdictions require you to pull a permit, even for DIY work of this nature.  Your electrician friend shouldn't mind as long as you pay the permit fees.  You get the benefit of having the work inspected, and the permit fee sometimes let the materials be purchased free of sales tax.  In the worst cases, without a permit, your insurance might not pay off, or you might not be able to sell your home in the future, or an overzealous code-enforcement department might even try to condemn your property.

There may be restrictions on where a 240V outlet can be placed in a residence.  [The kids' bedroom might not be a good idea.]  And unlike 120V circuits, which can feed multiple outlets, a 240V circuit may be limited to a single outlet.  [A 20A circuit will be overkill for one computer with a 5A power supply input.]

You do understand your savings won't be half?  Power = Voltage * Current.  So twice the voltage and half the current results in the same (approximate) power consumption.

As others have said, some of your savings will be in lower losses from your electrical panel to the back of the computer.  If you really want to be as energy efficient as possible, consider asking the electrician to use the next smaller gauge (larger diameter) compared to what code requires.  The wire will be more expensive, but you might make that back over a few years.  You could get some of this benefit by sticking with 120V circuits, but rewiring from 14 gauge to 12 gauge, for example.

The label of an HP power supply I am looking at says the input is 10A at 100-127V, and 4A at 200-240V.  That suggests the power supply may be more efficient at 240V.  Your power supply labels don't suggest a big difference in efficiency, but the manufacturer's Web Site may have better specs that would tell you if there is better efficiency at the higher voltage.

Finally, do you have a 240V electric dryer or electric range outlet in your home?  If you can find a cord, you could test before you rewire.  First, if you are concerned about burning out your computer, remove all of the expensive GPU cards and drives and test with the most bare-bones system you can.  I don't know if there is a 240V Kill-A-Watt meter in the US.  So, when you decide it works, turn everything in the house off, including the refrigerator.  Go out and see if the meter has stopped spinning.  Then turn on one computer, plug it in to 120V and fire up your miners.  Go out and watch the meter.  Time how long it takes to go around once (or a few times).  Then plug it in to 240V and fire up your miners.  Go out and watch your meter, again.  You probably can measure a 5% change in billed power consumption.