]I am certainly not a qualified electrician. But I do have a couple degrees in Electrical Engineering, and besides, it is not rocket science. A lamp is a bulb. Problem is to get electricity to bulb from mains. Most problems with lamps arise from loose connections along the way, or from the lack of
mechanical strength. Mum picks up lamp to dust it. Joggles a poor connection. Lamp does not work any more. Mechanical failure, not electrical: bulb screws into socket. Socket connections were not strong. Lamp failure. Electrical connections must be mum-proof. It is very difficult to get a shock off a modern lamp, but I suppose it is possible. Failure much more probable than shock.
UK and US have three-wire systems. One wire
neutral, one wire
live, one wire
ground. Oops, this is called
earth in the UK. In principle neutral is same as ground, in practice it may not be so. Voltage different, otherwise the same. US bulb sockets have but two terminals.One is live, the other neutral. I do not know how UK lamp sockets are wired. Principle is the same, however. If live ever touches either neutral or ground there is a pffft sound, a spark and probably a blown breaker. If you happen to be in the middle of live and either ground or neutral you will have at best an unpleasant experience, at worst it will be fatal.
US practice: connect neutral to one side of socket. Connect live to other side. There is a code in the US, live to "silver side", neutral to "brass side" -- refers to color of the screws. Disregard ground wire in lamp wiring. UK practice (and legal code) may be different -- never even thought of it, except to note that the plugs are shaped differently. For all I know you are to connect ground, not neutral, to t'other side from live, or there may be a ground connection on the lamp socket (a good idea, by the way). Read up on it in some DIY book. I always assume that a plug socket has been bodged, in the standard sense of the term; I have a neon tester that I use to find out if this is the case; $2 at any hardware store. Saves unpleasant shocks. So make sure mum's plugs are kosher before you give her the lamp.
Whatever you do, learn to connect stranded wire to a screw socket. I was taught to do this by a pro, age 15. It again is not rocket science. Requires a pair of long-nose pliers and a screwdriver. Make loop in wire, put around screw, tighten. Now tug the connection,
hard. If it comes off you did it wrong. Play it again, Sam

. And if you do use a tube to lead the wire to the socket, US practice is to tie an overhand knot in the cord at both ends of the tube. This is to prevent jerks from loosening your connections. Amazing to me how many people unplug a lamp by jerking the cord.