Since there is a cup product map $H^1(X;\mathbb Z) \times H^2(X;\mathbb Z) \to H^3(X;\mathbb Z)$, one should be able to produce from a principal $\mathbb Z$-bundle and a principal $S^1$-bundle a principal $PU(\mathcal H)$-bundle, where $PU(\mathcal H)$ is the projective unitary group of an infinite dimensional separable Hilbert space $\mathcal H$. This is because $BS^1 \simeq K(\mathbb Z,2)$ and $BPU(\mathcal H) \simeq K(\mathbb Z, 3)$.
Indeed, using these classifying spaces and the cup product map on Eilenberg-Maclane spaces, $K(\mathbb Z, 1) \times K(\mathbb Z, 2) \to K(\mathbb Z, 3)$, one can get the classifying map for the $PU(\mathcal H)$-bundle (although I don't know how to explicitly write down what this map is). I would like to know whether anyone is aware of a concrete and direct construction using the principal bundles themselves?