By induction on $2 \leq \delta=$length of (u)$< \kappa,$ one can show that $otp(C)= \omega^{\delta-1}$, if $\delta < \omega$ and $otp(C)= \omega^{\delta}$ if $\omega \leq \delta < \kappa$ where $C$ is the Radin club, in particular if $\delta$ is regular uncountable, then $otp(C)=\delta$. Further one can manage to not add any new subsets to $\delta$. Now the result should be clear.
Here is the basic idea:
The case $l(u)=2$ is just the case of Prikry forcing, and we have $otp(C)=\omega.$
Suppose $\delta=\gamma+1$ is a successor ordinal and it holds for $\gamma$. Then note that forcing with $\mathbb{P}_u$ adds an $\omega$-sequence $(w_i: i < \omega)$ of measure sequences each of them of length $\gamma.$ Now use the factorization of the forcing and the induction hypothesis to get the result
for $\delta$.
For limit $\delta < \kappa,$ consider the set $A=\{ w \in U_\infty: l(w) < \delta \}\in \bigcap_{0<\alpha < \delta}u(\alpha)$, and show, using the induction hypothesis, that below $(u, A),$ the club filter $C$ is forced to have order type $\omega^\delta=sup_{i<\delta}\omega^i.$
Radin forcing changes cofinalities below $\kappa:$
Radin forcing changes cofinalities below $\kappa$ too, provided that $length (u)>2$. This is clear, because of the following: Let $(u_i: i<\theta)$
be the measure sequence added by generic filter such that the sequence $(\kappa_i=\kappa(u_i): i < \theta)$ is increasing. Recall that $C=\{ \kappa_i: i \}$ is the Radin club. We can easily manage all $\kappa_i$'s be measurable in $V,$ the ground model: just note that the set $A'=\{w \in U_\infty \cap V_{\kappa(u)}: \kappa(w)
$ is measurable$ \}\in u(\alpha), 0 < \alpha < length (u)$, and it suffices to force below $(u, A')$.
As $length (u)> 2,$ so $\theta=otp(C) \geq \omega^2$, in particular all $\kappa_i, i \leq \omega^2$ are singular in the extension with cofinality $\omega$.
In fact it is possible to say much more: First assume $length (u)< \kappa$, so $\theta=\omega^{lenght(u)}$. Then we can also manage $\kappa_0=min(C)> \theta,$ so clearly each limit point of the Radin club is singular in the extension.
Again more is possible to say, if $length (u)=\kappa^+,$ then $C$ has order type $\kappa,$ and by a theorem of Mitchell, $\kappa$ remains inaccessible in the extension, but if we force below $(u, A''),$ where $A''=\{ w \in U_\infty \cap V_{\kappa(u)}: length(w) < \kappa(w)^+ \}$, then all limit points of the Radin club will be singular in the extension, so that $\kappa$ becomes inaccessible not Mahlo in the extension.
