= Solution
We use the <core-quotient bijection for partitions>: an ordered $p$-tuple $(\nu_0,\ldots,\nu_{p-1})$ and a fixed $p$-core determine a unique <partition of an integer>, with
$$
|\lambda|=|C_p(\lambda)|+p\sum_i|\nu_i|.
$$
The <irreducible characters> in both $B$ and $C$ therefore correspond to the same ordered tuples of total size $w$. Replacing the empty <core of a partition> by $\gamma$ while keeping the <quotient of a partition> fixed gives a <bijection> $\chi^\lambda\mapsto\chi^\mu$ from $B$ to $C$. In particular $|B|=|C|$.
For such a pair, all levels of the <core tower of a partition> above level zero agree. At level zero, their sizes are $0$ and $m$, respectively. Put $D=\sum_{r\geq1}|T^C(\lambda)_r|=\sum_{r\geq1}|T^C(\mu)_r|$. When $0\leq m<p$, the <base-p digit sum> satisfies
$$
d_p(wp)=d_p(w),\qquad d_p(wp+m)=d_p(w)+m.
$$
The <P-adic valuation of a symmetric-group character degree from the core tower> then gives
$$
\nu_p(\chi^\lambda(1))=\frac{D-d_p(w)}{p-1}
=\frac{m+D-[d_p(w)+m]}{p-1}
=\nu_p(\chi^\mu(1)).
$$
Thus the same bijection preserves degree coprimality to $p$ and restricts to a bijection between the indicated sets of <irreducible characters of degree coprime to p>. Consequently
$$
\boxed{|B|=|C|,\qquad
|B\cap\operatorname{Irr}_{p'}(S_{wp})|=|C\cap\operatorname{Irr}_{p'}(S_n)|.}
$$
For $m=0$, take $\gamma$ to be the empty partition; the construction becomes the identity.
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