For n=2 use the p-adic expansion of $\zeta$ to express an explicit uniformizer: combinations of $(\zeta-1)$ and $p^{1/p^2}$ continue canceling terms in the expansion until the first term with p-exponent having denominator divisible by $p^3$. For example with p=2: $(\zeta-1)2^{-3/4}+2^{-1/4}+1 = 2^{1/8}+...$ is a uniformizer.

For n=2 use the p-adic expansion of $\zeta$ to express an explicit uniformizer: combinations of $(\zeta-1)$ and $p^{1/p^2}$ continue canceling terms in the expansion until the first term with p-exponent having denominator divisible by $p^3$. For example with p=2: $(\zeta-1)2^{-3/4}+2^{-1/4}+1 = 2^{1/8}+...$ is a uniformizer.

Edit: With p=3: $\zeta_9 = 1 + \zeta_4\times3^{1/6} + 3^{1/3} - 3^{4/9} + 3^{13/27} ...$ and so $3^{2/3}\times((\zeta_9 - 1 - 3^{1/3} + 3^{4/9})^2 + 3^{1/3})^{-1} = -\zeta_4/2\times3^{1/54}+ ...$ is a uniformizer.