  Let $ R_\gamma B^{s}_{p,q}(\mathbb{R}d)$ be a subspace of the Besov space
$B^{s}_{p,q}(\mathbb{R}^d)$ that consists of block-radial (multi-radial)
functions. We study an asymptotic behaviour of approximation numbers of compact
embeddings $id: R_\gamma B^{s_1}_{p_1,q_1}(\mathbb{R}^d) \rightarrow R_\gamma
B^{s_2}_{p_2,q_2}(\mathbb{R}^d)$. Moreover we find the sufficient and necessary
condition for nuclearity of the above embeddings. Analogous results are proved
for fractional Sobolev spaces $R_\gamma H^s_p(\mathbb{R}^d)$.
