>>1670920tl;dr wider rim = wider tire when inflated, but different brands might have different sizes regardless of what it says on the label.
d = r(2pi-2t)
w = 2r sin(t)
Eliminating t gives
d = r (2pi-arcsin(w/2r))
This is the relation between tire casing width ('d'), rim width ('w') and inflated radius ('r'). To calculate the tire casing width either measure it directly or calculate it by measuring the inflated diameter on your current wheels with calipers (or an adjustable wrench) then dividing by 2 and using the formula. Once you have d, plug in the new rim width w into the equation and solve for the new r with wolframalpha.