  We perform the 4-dimensional Kaluza-Klein (KK) reduction of the 5-dimensional
locally scale invariant Weyl-Dirac gravity. While compactification unavoidably
introduces an explicit length scale into the theory, it does it in such a way
that the KK radius can be integrated out from the low energy regime, leaving
the KK vacuum to still enjoy local scale invariance at the classical level.
Imitating a $U(1)\times\tilde{U}(1)$ gauge theory, the emerging 4D theory is
characterized by a kinetic Maxwell-Weyl mixing whose diagonalization procedure
is carried out in detail. In particular, we identify the unique linear
combination which defines the 4D Weyl vector, and fully classify the 4D scalar
sector. The later consists of (using Weyl language) a co-scalar and two
in-scalars. The analysis is performed for a general KK $m$-ansatz, parametrized
by the power $m$ of the scalar field which factorizes the 4D metric. The
no-ghost requirement, for example, is met provided $-\frac{1}{2}\leq m \leq 0$.
An $m$-dependent dictionary is then established between the original 5D
Brans-Dicke parameter $\omega_5$ and the resulting 4D $\omega_4$. The critical
$\omega_5=-\frac{4}{3}$ is consistently mapped into critical $\omega_4 =
-\frac{3}{2}$. The KK reduced Maxwell-Weyl kinetic mixing cannot be scaled away
as it is mediated by a 4D in-scalar (residing within the 5D Weyl vector). The
mixing is explicitly demonstrated within the Einstein frame for the special
physically motivated choice of $m=-\frac{1}{3}$. For instance, a super critical
Brans-Dicke parameter induces a tiny positive contribution to the original (if
introduced via the 5-dimensional scalar potential) cosmological constant.
Finally, some no-scale quantum cosmological aspects are studied at the
universal mini-superspace level.
