It should - the Mk7/7.5 (which the US models are) will still fit 15" wheels wheels over the stock front brakes, and as far as I can tell, the only difference with the Mk8 front brakes is a slightly different caliper on the same rotor size. I currently run that same winter size (195/55R16) on my US/Mk7.5 with no rubbing (at stock ride height though, not lowered). The Mk8 has an extra half inch of tire well clearance with the 18" wheels, so you're likely to be just fine with the tire diameter as well.
Check the size of your spare tire in the back - I'm assuming the Mk8's have one. Dollars (Zloty?) to donuts it still uses the same 15" spare tire size as the previous generation Fiestas. If so, your winter tires will definitely fit.