Mining is not needed to craft, but IS needed to recycle items you make, but don't want.
Your ingot return for recycling items is 0.66% * Mining skill, and you can't recycle items of metals you can't mine.
It would probably be easier just to leave your current mining on your smith, and train mining up on the new character. You can get it to 50 in a matter of a couple minutes in New Haven (and get a pickaxe as a quest reward that will give you +10 mining skill when held - equip BEFORE putting mining gloves on, to combine their bonuses, but you have to use it to mine when holding it). From there, just going out and mining iron will get you into the range for colored ore within a couple hours, and be fully trained after a day or two of mining (just be sure to take a fire beetle). That quest reward pickaxe only has 20 uses, but doesn't break from running out of charges, instead recharging about a use every 3 minutes (even in your pack)