Relations between characteristic classes of a group and the Stiefel-Whitney/Pontryagin classes Let $X$ be a closed manifold and $BG$ be the classifying space of a group $G$
A map from $X$ to $BG$ induce a map from $H^*(BG,Z)$ to $H^*(X,Z)$ by pull back.
Let $GH^*(X,Z)$ be the subgroup of $H^*(X,Z)$ formed by the images of the above map
$H^*(BG,Z)\to H^*(X,Z)$ for all the maps $X\to BG$.
In this case $a \in GH^*(X,Z)$ and the Stiefel-Whitney/Pontryagin classes may have some relations that are indenependent of $X$. What are those relations?
If we choose $G=U(1)$, this question bacomes: what are the relations (all the $X$ independent relations) between the Chern classes of a $U(1)$ bundle on $X$ and the Stiefel-Whitney/Pontryagin classes on $X$.
In question Relations between Stiefel-Whitney classes, the relations between Stiefel-Whitney classes on any $X$ are discussed.
== Added ==
This may be the better way to phrase the question:
What are relations between the characteristic classes of the tangent bundle of a manifold and characteristic classes of an arbitrary $G$-bundles on the same manifold. 
I think in 4-dimension and for $G=U(1)$, one of the relation is $(w_2+w_1^2) c^{U(1)}_1 = c^{U(1)}_1c^{U(1)}_1$ mod 2.
 A: There's no relations between SW, Pontrjagin, and Chern classes (within each set). On the other hand, for a $U(1)$-bundle, one has $w_1=0$ and $w_2=c_1\bmod2$. Such a bundle has no (rational) Pontrjagin classes.
More general setting
Here is another bunch of relations; most likely these are all, but I'm not 100% certain. Let $\dim X=n$, and let $u_i:=u_i(X)$ be the $i$-th Wu class of $X$. (Recall that the total class is $u=\operatorname{Sq}^{-1}w(X)$, and $u_i$ are the homogeneous components.) Then, for any $a\in H^{n-i}(X;\Bbb Z_2)$, one has $u_ia=\operatorname{Sq}^ia$ in $H^n(X)=\Bbb Z_2$. This gives us relations $u_i=0$ for $i>n$, and these generate the ideal of all relations among the SW classes $w_*(X)$, see Relations between Stiefel-Whitney classes. Now, if one takes for $a_j\in H^j(X;\Bbb Z_2)$ the characteristic classes (SW, Chern, Pontrjagin, or such) of another bundle (or combinations thereof), then, using Wu formulas to express $\operatorname{Sq}^{n-j}a_j$ in terms of the $a_k$'s and substituting to the above formulas, one gets a bunch of relations between $w_*(X)$ and $a_*$.
The relation $(w_1^2(X)+w_2(X)) v_2=v_2^2$ mentioned in the comments is one of them. (Here, $v_2\in H^2(X;\Bbb Z_2)$ is any class, which, if desired, can be interpreted as $w_2$ or $c_1$ or whatever of another bundle.) Another one for a $4$-manifold is $w_1(X)v_3=v_1v_3$, where $v_i$ are the SW classes of another bundle. 
