So I have all the equipment made and now I want my dwarves to change their crappy copper picks for the better iron ones. I also want my wood cutters to use the better axes, but to no avail. Please help me, or I'll be stuck in the weak age for ever :(
In terms of job performance, material has no effect on the Dwarf's ability to perform the task, though there is something to be said of providing better tools for when the Woodcutter or Miner runs afoul of the rogue carp.
You've got a couple ways to do it, but the fastest would be to forbid all the non-iron picks via the stock screen (you may need a bookkeeper if you have more than 10). Your Miners will then drop the forbidden tools and ferret out your iron picks instead. Unforbid, melt down, or otherwise handle the copper tools as you will, or the next dwarf you promote to miner might take up the pick as well.