Here are two papers that do not allow to conclude positively to your question. The reason is that one does not know how to bound uniformly the Mordell-Weil rank of abelian varieties of given dimension over a given number fields, even in well behaved families.
1. A general bound depending on the size of the coefficients
In his paper ``Borne polynomiale pour le nombre de points rationnels des courbes'', Journal de théorie des nombre de Bordeaux 23 no. 1 (2011), p. 251-255, Gaël Rémond has given a general explicit bound for the number of solutions of polynomial equations in two variables with coefficients in a number field, assuming the equation has finitely many solutions, of course. By Faltings's theorem, this is the case of your curve, so his bound applies and says that there are at most $n^{2^{3^{16}}}$ solutions.
NB. The exponent of $n$ is equal to $1.721783764\, 10^{12958354}$ and this bound is both unpractical and certainly non-optimal.
NB. Rémond bounds the Mordell-Weil rank in terms of the size of the equation.
2. A bound for twists of a given curve
The paper A uniform bound for rational points on twists of a given curve, J. London Math. Soc. (2) 47 (1993) 385-394, by Joseph Silverman shows a partial uniformity of the number of rational points among twists of a given curve. However, the bound depends on the Mordell-Weil rank of the specific curve, so is not really uniform.
He also gives the example of Catalan curves of the form $ax^m+bx^n=1$, where $m$ and $n$ are fixed and $a,b$ varies among non-zero rational numbers.