A useful reference on this topic (which maybe you know?) is "The Manin Constant",
by Agashe, Ribet, and Stein, available here. On p. 3 they write

B. Edixhoven also has unpublished results (see [Edi89]) which assert that the
only primes that can divide $c_E$ are 2, 3, 5, and 7; he also gives bounds that are
independent of $E$ on the valuations of $c_E$ at 2, 3, 5, and 7. His arguments rely
on the construction of certain stable integral models for $X_0(p^2 )$.

The reference [Edi89] is to Edixhoven's thesis, which is unpublished, but
which can be found here
at Edixhoven's website. At the end of the introduction to his thesis, Edixhoven writes

Finally,
in the last section, we derive some results concerning the constant
"c" attached to a strong Weil curve $E$. Manin conjectured that $c=1$. It
is known that $c$ is a positive integer and Mazur proved that only 2 and
primes where $E$ has additive reduction can divide $c$. Our methods show
that primes $p>7$ where $E$ has additive reduction divide $c$ at most once,
and in fact, for most of the possible reduction types (=Kodaira symbols), not (Theorem 4.6.3 and the remarks following this theorem). It might well
be that a computation involving the period lattices of normalized newforms
can solve the problem in the case of potentially good, ordinary reduction of type $II, III$ or $IV$. It should also be tried to get bounds on the exponents
of 2, 3, 5 and 7 in $c$.

This suggests that the statement of Agashe, Ribet, and Stein is perhaps a little strong,
in that Edixhoven does not give uniform bounds at all primes, but only at primes $p > 7$,
with the suggestion that one could also hope to obtain such bounds at 2, 3, 5, and 7.
You might try writing to Edixhoven for clarification, or perhaps also to one of Agashe, Ribet, or Stein to find out more precisely what they had in mind.