It depends on coordinates
General Relatively says that time is inexorably linked with space (they bend alongside each other).
This is not generally (pun intended) true; it sounds like something out of pop-science, which for your edification I recommend you disregard. In the FLRW model of expanding spacetime only the spatial part of spacetime is actually expanding. It looks like
$$c^2\text d\tau^2=c^2\text dt^2-a^2(t)(\text dx^2+\text dy^2+\text dz^2)$$
for a Euclidean spatial part (elliptical/hyperbolic are also options). So as $a(t)$ increases over time, the time dimension remains constant. It is not curved in the same sense that the $x$ and $y$ and $z$ dimensions are. Here the speed of a null $x$-directed ray $\frac{\text dx}{\text dt}\big|_{\text d\tau=0}$ is $c/a(t)$.
That being said there is a conformal transformation (a certain type of geometry-preserving transformation) that just affects time. Set $\Omega(x^\mu)=\frac{1}{a(t)}$, then transforming $g_{\mu\nu}\to g'_{\mu\nu}=\Omega^2(x^\mu)g_{\mu\nu}$ gives
$$c^2\text d\tau^2=\frac{c^2}{a^2(t)}\text dt'^2-\text dx'^2-\text dy'^2-\text dz'^2$$
where only the time dimension is curved. Here the speed of a null $x$-directed ray $\frac{\text dx}{\text dt}\big|_{\text d\tau=0}$ is still $c/a(t)$, so the net effect is still the same.
