No, No, No. Modern appliances with exposed metal parts will be
connected to Earth.

We're not talking about a modern appliance, is the problem. We're
talking about how to make an antique a little safer.

This means it should be in contact with the EARTH pin. NEVER connect
exposed metal parts to neutral.

You can if you like. It will trip your RCD (residual current device),
if you have one, but not everybody does. These thingsa are called GFI
(ground fault interruptor) devices in the US. But if you don't have one
of these in your house, or on the circuit in question, then it does no
damage for neutral to touch ground in the lamp or whatever. And it may
make the lamp safer.

It may appear that the neutral wire has no voltage but this is not
so.

That depends. Actually, it is so, UNLESS you have an RCD/GFI, which
places a small voltage between neutral and ground as they exit the RCD
device. After all, the ground and neutral are connected together in the
service panel, so there's no difference between them unless the RCD,
connected after the panel, puts it there. The purpose of this small
voltage is so the RCD can detect a ground-neutral "fault" (one where
the neural wire has touched the gound inside your lamp or other
appliance) and trip when that happens. The reason RCDs do this is
because having the neutral touch ground makes the major feature of the
RCD (the ability to tell if more current is flowing in the hot wire
than in the neutral, signaling an extra path to ground which may be you
being electrocuted) inoperative. So the RCD is designed to shut off the
power if its sensing function (which requires a separate neutral and
ground path as they exit the device) is compromised and it becomes
"blind" and unable to perform its function. But again, if you don't
have an RCD in the first place, none of this applies. YOU don't care if
the ground touches neural in your old lamp UNLESS you have an RCD/GFI
to trip in the first place.

If a short is placed between earth and neutral the RCD in your
consumer unit will trip.

Yes, but again, if you don't have an RCD, it won't. So you don't care.
If you don't have an RCD it's better to have a grounded device outer
casing, at least, rather than have one that floats. The latter is a
shock hazard if internally it should touch the wrong wire and become
hot. But if the casing is connected to neutral at least, then you have
a chance that a hot wire contacting it will trip your fusebox, which is
a poor man's GFI. All this assumes, of course, that you have only 2
wire plugs, which are polarized. These still exist many places in the
US. I was assuming that this antique lamp had this design, but it
doesn't sound like it. If not, then obviously the 3rd (ground/earth)
wire must be followed to make sure it contacts the lamp casing instead.

If it doesn't there is an earth fault. Get it seen to.

You say, if a short between earth and neutral doesn't make your RCD
trip then there is an Earth fault??? Sorry, that doesn't make sense.
If it doesn't, then it means your RCD isn't working.

Anyway, this guy sounds like he has an old lamp with a polarized 2 pin
plug, and no ground pin. The lamp is apparently designed with a
floating casing, so that neigher hot or neutral connect to the metal
arms of the lamp. Do I have the correct impression?

SBH