Ipod won't start

I'm not sure which on/off switch you are referring to (iPod, or (in the UK), the wall switch), but I would expect the following::

• if battery is full - this appears unlikely in your case, but I would expect the iPod to show the battery for a few seconds when you first plug it into a power source
• if the battery is less than full - since it can accept charge, I would expect the display to show the battery charging icon for a few seconds after first plugging it in. Subsequently, pressing the iPod's power button should case the display to turn on for a few seconds - but note: when you first plug the iPod in, the display should turn on without you touching any of the iPod's buttons
• if the battery is fully drained (which as discussed, appears to be your situation) - no reaction at all from the iPod for anything up to thirty minutes. Then the display should turn on briefly. A bit later, it should go through its start up procedure

Something that we haven't mentioned:

• are you sure the cable from the charger to the iPod is good? Have you tried another cable (borrow a sibling's, or a friend's cable)?
• Have you tried the cable with a known working iPod?
• The same applies to the wall charger you mentioned
• In the USA (where I believe every wall outlet has its own breaker elsewhere in the house), have you checked to ensure that the wall outlet is working?

If the battery is fully discharged it can take up to thirty minutes after plugging into a power source before you see any reaction from the iPod.

Have you allowed for this? If not, try it.