I'm following a solution of an SDE from here

http://www.math.ethz.ch/~delbaen/ftp/preprints/CEV.pdf

Start with the SDE $$ dX_t = \delta dt + 2\sqrt{X_t} dW_t $$ consider a deterministic time change $$ \tau = \frac{\sigma^2}{2\nu(2-\delta)}\left(1-\exp\left(-\frac{2\nu t}{2-\delta}\right)\right) $$ the process $Y_t$ is defined as $$ Y_t = exp(\nu t) X_{\tau}^{1-\delta/2} $$ then, using Ito's lemma, we get $$ dY_t = \nu Y_t dt + \sigma Y_t^{\frac{1-\delta}{2-\delta}} dW_t $$

I'd like to understand how the Ito's lemma is applied here. The problem that I have is that when you write it, the $Y$ is derivated with respect to the $X$, but in the definition of $Y_t$ is used $\tau$, not $t$, in $X$, so there's a mixture between different times. How is this handled?