Let’s go back to when electricians were poor and couldn’t have the MFT tester (not invented) and the trade wasn’t a restricted trade.
when I was 5 this was the test that would be done.
consider the size of the cable (not bell wire )
Buy a cheap multimeter - useful for checking fuses. I had access to an AVO meter.
now:
Unplug lamp and Remove bulb from lamp.
The plug was 3 pins.
The lamp holder has two pins and the metal case.
If the lamp has a switch switch it to the on position.
Check continuity from each of the pins on plug with pins on the lampholder.
There should be one and only one match for each set of pins.
Now put the switch off.
At least one of the continuity passes should not work.
Now switch on the switch again and plug in the bulb.
Test the continuity between the two pins on the plug ( not the earth) and you should get a small reading but not 0.
this test is not sufficient for selling the lamp but should be sufficient for your use ( it was for a large number of years)
This sounds like a lot but it would take 3 minutes.
hope this helps
I disagree. A lamp that has been rewired by an amateur should be tested properly. Personal use doesn't come into it - are you saying it's OK that his family may receive an electric shock?