To get the composition F(G(x,y)):
Take your expression for F in terms of u and v. These are just placeholders. Set (u,v) = G(x,y) = (x^2 + 1, y^2). Now everywhere where you see a "u" in the expression for F, substitute "x^2 + 1" and likewise, everywhere where you see a "v" in F(u,v), substitute "y^2".
Therefore F(G(x,y)) = (x^2 + y^2 + 1, (y^2)^2 )
= (x^2 + y^2 + 1, y^4 ). Now you can calculate the Jacobian of that in terms of x and y.