Here is a simplified version of AC power lines:
simulate this circuit – Schematic created using CircuitLab
The two lines at lower right represent you holding the neutral. Since the neutral is connected to ground elsewhere, as you agree you only feel a small voltage. Note that the load is fairly low resistance, so it can produce a lot of power.
Now let's break the neutral
simulate this circuit
Current will now flow through the upper hot wire, through the load, through you and then the ground in order to get back to the transformer neutral. And since the load has a low resistance, you are the big resistor in the circuit, and you will take most of the voltage.
A couple of things.
A very small point, the UK is in Europe, and AFAICR we have had 3-pin plugs, 3-wire cables, since the late 50's.
My house was rewired in the 80's, and we have Earth Leakage Circuit Breakers (ELCB) on every circuit. So even if I stuck a fork into my toaster while grabbing a copper water pipe, I'd expect it to trip (I am not willing to back this up with evidence :-)
When I have visited continental Europe, I am pretty sure that I've seen the same ELCB technology in use.
I suggest that is even more effective than having an Earth connection; after all, if I touched the correct bit of wire in your toaster with my fork, without touching anything else, the Earth connection via a plug would do me no good. Further, unless the device had a metal case connected to Earth, I don't think I am much more likely to touch both Earth and live than just live alone.
I imagine the cost of rewiring all of the houses in Europe which have two-wire cabling t have three-wire would be very large. However, upgrading the distribution panel with ELCB is pretty simple (a drop in replacement in some cases for an old fashioned fused unit), and could be caused to happen more easily when electricity metres need replacing.
Best Answer
The neutral is NOT safe to touch.
When everything is working correctly, it should be at most a few volts from ground. However, and this is the big gotcha, if there is a break in the neutral line between where you are and where it is connected back to ground, it can be driven to the full line voltage. Basically in that case you are connected to the hot line via any appliances that happen to be on in that part of the circuit. Those can easily pass the few mA it takes to kill you.
This is not supposed to happen, but since failures can be lethal and costly, a extra layer of safety is built into the protocol and rules. It is irresponsible and needlessly risky to consider the neutral line safe.
This is why modern appliances either have two prongs and everything is insulated from the user, or three prongs and anything conductive the user can touch is connected to ground. In some past cases there have been appliances with polarized 2-prong plugs, but those are seriously frowned upon today. I don't think you're going to get UL approval for such a device unless it is fully insulated, in which case you shouldn't need a polarized plug.