There cannot be such a uniform bound, unless $p=2$ in which case there are
no solutions at all (because $p \notin S$ but $u$ and $1-u$ cannot both be 
$2$-adic units).

Fix $p$ and $e$.  We shall construct a set $S$ of uniformly bounded size
and $S$-units $u_1,u_2$ that are congruent modulo $p^e$.

By a theorem of Chen (*Sci. Sinica* **16** (1973), 157-176),
there are infinitely many primes $q$ such that $q-2$ is 
either a prime or the product of two primes (and as usual this remains 
true if we require that neither $q$ nor $q-2$ is a multiple of $p$). 
Hence $q/2$ is an $S(q)$-unit for some set $S(q)$ of at most $4$ primes.
Therefore there exist distinct $q_1,q_2$ satisfying this condition
with $q_1 \equiv q_2 \bmod p^e$.
Then $u_1 := q_1/2$ and $u_2 := q_2/2$ are both $S$-units for some 
$S = S(q_1) \cup S(q_2)$ of size at most $8$ 
(indeed at most $7$ because $S(q_1),S(q_2)$ both contain $2$);
and $|u_1 - u_2|_p < p^{-e}$.  $\Box$