Using the relay to disconnect the neutral line doesn't really sound like you know what you're doing. You're exactly wrong in the safety department. In a perfect world, it's the hot that you want to switch. With only neutral switched, the heating coil is always energized, waiting to shock you.
Additionally, relying on a polarized electrical cord to try to make sure it's the hot you're switching has its own problems. You're better off using a double pole relay to fully disconnect the toaster from the electrical supply. Use a relay that is designed specifically to switch both legs of a power supply.
Also don't need to heat-shrink, glue, or electrical tape the wires, that's overkill and amateur hour. Hiding things in an attempt to increase safety is a sure way to decrease safety.
Use the project box, and mount your circuitry (including relay) on perf board. Simply solder the relay inline on the hot wire. Keep your low voltage relay control wiring (and arduino circuitry) physically separate from the relay and line voltage power wires. Do not let low voltage and high voltage come near each other, except at the relay, and even then, those come from different directions. Physically secure the power wires so they can't accidentally be pulled out. Be neat.
Your heating coils should be shielded from your pcb by a heat spreader, and your thermocouple should be next to the pcb, and not be touching anything. It's the air that heats up the pcb, and the air that heats up the thermocouple. You trying to bake the pcb, not broil it.
IMPORTANT SAFETY NOTE:
The electrical power that comes out of your outlets is extremely dangerous. It's not the fun-times easygoing stuff of 15 volts and below. It will surprise you, and given the opportunity, it will kill you.
From your question, you really don't seem to know what you're doing. I'm not trying to insult you, I just don't want you to die. Advice from the internet is not going to make this any safer. Sit down with someone in the real world who has worked PROFESSIONALLY with household electrical circuits and have them look over what you are doing.
Please be safe.
As other posters have suggested, this can't be done via conductive measurement alone. However, non-contact AC testers will properly differentiate line from neutral without a reference ground via capacitive sensing. This type of hand-held tester often has a flat plastic blade at the tip. Inserting the blade in the line slot will cause the LED to glow; inserting in neutral or ground will not glow:
There are a number of kits for this type of circuit, as we as many sample schematics available in a few minutes of searching (Search string: non-contact ac voltage detector circuit). They're essentially just high gain amplifiers that drive an LED.
Using this type of sensor permanently mounted near the Neutral line can serve as an indicator that your plug has been inserted the wrong way, provided that you have a reliable way (i.e. not battery) of powering the circuit.
Best Answer
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.