GCD of a pair of products
You can compute the pairwise GCDs of the factors. You have to divide out the GCDs from the factors to avoid finding the same GCD factor twice:

a[1...m] = [given factors of A]
b[1...n] = [given factors of B]
g = 1
for i from 1 to m do
    c = a[i]
    for j from 1 to n do
        d = gcd(c, b[j])
        g = g * d
        c = a[i] / d
        b[j] = b[j] / d
return g


Unfortunately, I don't think that this is not notably faster than the computation of the GCD of the two numbers A and B.