On common algebraic points of a pair of different derivatives of finite-order meromorphic functions

Given nonnegative integers $t<s$, pairs ($f^{(t)}(z)$, $f^{(s)}(z)$) of derivatives of finite-order meromorphic functions $f(z)$ for which $f^{(t)}(z)$ is neither a rational function, a rational function of an exponential $e^{\alpha z}$, nor an elliptic function are considered. For positive integers $n$ and $H$ and a positive number $R$, let $B(n,H,R)$ be the set of points $z$ in the disk $|z|\le R$ for which $f^{(t)}(z)$ and $f^{(s)}(z)$ are algebraic numbers of degree at most $n$ and height at most $H$ (and, moreover, $|f^{(t)}(z)|$ and $|f^{(s)}(z)|$ are not very large). An upper bound for the number of points in $B(n,H,R)$ is obtained for almost all, in a certain sense, $(n,H,R)$.