Gluing (Principal) Bundles?

Question

Say $X $ is a manifold, and let $U$ and $V$ be two disjoint submanifolds. Then given two Lie groups $H_1$ and $H_2$ we can fiber $U$ with $H_1$ fibers and similarly with $V$ and $H_2$. It seems to me that one can "lift" the fibration to one of $X$ in a naive way: pick a group $G$ which contains both of the $H_i$ groups, fiber $G$ over $X$ such that the induced injections into the submanifolds map into the original fibrations by gluing cosets. This seems plausibly kosher by adding a few lines of math, even if the lift is far from unique. My question is the reverse of this, given two principal bundles is there a nice (canonical) way of gluing these two together?