failure of Projective Uniformization How do you show the failure of Projective Uniformization in the generic extension of $L$ which adds $\aleph_{2}$ Cohen reals?
 A: Consider $A=\{(x,y)\in\mathbb R^2\mid x\notin L[y]\}$. 
Check that this set is $\Pi^1_2$ (this is similar to the proof that there is a $\Delta^1_2$ well-ordering in $L$). 
The point is that $A$ does not admit a projective uniformization. It does not really matter that the number of Cohen reals you added is $\aleph_2$; any uncountable number would work. The reason is essentially the same as the reason that choice fails in $L(\mathbb R)$ of the forcing extension (a consequence of the homogeneity of Cohen forcing and the fact that any real parameters used in a projective definition of a set are added by an intermediate countable poset; this allows you to diagonalize and argue by contradiction) so not only there is no projective uniformization of $A$, but not even a uniformization in $L(\mathbb R)$. 
By the way, this result was established quite soon after forcing was developed, see 

MR0205827 (34 #5653) Reviewed. Lévy, Azriel. Definability in axiomatic set theory. I. In Logic, methodology and philosophy of science. Proceedings of the 1964 International Congress, Y. Bar-Hillel, ed., North-Holland, Amsterdam, 1965, pp. 127–151.

