I unplugged the lamp and tested the resistence between the live
pin on the mains plug and some exposed metal on the lamp. I got
no resistence reading at all (i.e. it must have been a very high
resistence). I then tested the neutral pin in the same way and
got the same high resistence result.

Many appliences ground the outside of metal appliance, which means it
should be in electrical contact with your neutral pin. Which means THAT
resistance should have been LOW (zero). This is a measure to prevent
shock, in case something inside the appliance should contact that outer
case (which would then blow your fuse instead of making the case "hot",
and leave it as a safety hazard in case you should grab it in one hand,
and something else that goes to ground in the other).

If you can follow the grounded pin inside your lamp, you should see
that it ends SOMEWHERE. Probably it goes to the metal screw fitting of
the bulb. If you really want to make your lamp safer, you can run a
wire from that to the metal arms somewhere. Then you can't ever shock
yourself by grabbing it.

Caution--- it's only safe to do this if your 2-pole plug is polarized
so it can't be put into the wall, in reverse. I have no idea what UK
plugs look like, so you'll have to tell me if this is true. From your
description it sounds as though it is.

SBH