As Keith Conrad observed in his comment, what you are asking about is Washington's theorem which is either in his book, referenced as Theorem 16.12, or in his original paper in Invent. Math. He proved that for each abelian number field $K/\mathbb{Q}$ and every pair $\ell\neq p$ of odd primes (I am not sure at what happens with the prime $2$), the $\ell$-part of the class group in the cyclotomic $\mathbb{Z}_p$-extension of $K$ stays bounded. What is not known is if only finitely many $\ell$ can appear.
Horie has made this explicit in his two papers
- K. Horie, Ideal Class Groups of the Iwasawa-theoretical extensions over the rationals. J. London Math. Soc. 66 257–275 (2002)
- K. Horie, Triviality in ideal class groups of iwasawa-theoretical number fields. J. Math. Soc. Japan, 57 827–857 (2005)
The main difficulty, as you say, is that the ring $\mathbb{Z}_\ell[\![\mathbb{Z}_p]\!]$ is not so nice. But one can study representations of $\mathbb{Z}_p$ with values in larger and larger quotients $\mathrm{GL}_n(\mathbb{Z}/\ell^k)$ for $k\to\infty$ and still get relevant information.
