No, it won't. We've already proven that technologies can be adopted very quickly if you have an example provided. And considering the fact that every single hill civilization we have found has had metal in their hills, it's a reasonable bet that we do too.
In order to get useful amounts of copper, we need several things. First, we need to survey our hills and find sizeable deposits of copper. As you said, this shouldn't be too difficult. Not guaranteed, but still. It
will take an action, however, which we could be short on like we are now.
Then comes the difficult bit, which is setting up the mines. The Metal Workers are fine with trading people metal, but things like teaching people how to mine? They'd be breaking their own monopoly for people who are far away, not connected by trails, and have had relations with for just a few generations at most. Still even assuming they'd be willing to share their secrets, that's still actions we'd have to take in order to get a proper foundation of professional miners set up to actually do the mining.
We'll have to take even more actions to get a proper base of trained blacksmiths in order to work the metal, actions which we might have to use on dealing with natural disasters, or quickly beefing up our military to deal with increased DP pressure against us, or dealing with stability issues, or any number of other possibilities.
So no, while quickly getting mining/metalworking is possible
in theory, it's very unlikely that that'll be the outcome with all the variables in play and assumptions we'd have to make.