The paper

Harant, Jochen; Rautenbach, Dieter; Recht, Peter; Regen, Friedrich. Packing edge-disjoint cycles in graphs and the cyclomatic number. Discrete Math. 310 (2010), no. 9, 1456--1462

constructs graphs for which the difference between $d$ and the nmaximum number of edge-disjoint cycles is $k$ for any $k \in \mathbb{N}$. If I understand the question correctly this shows you can't have a bound desired without some restrictions on the graphs (I have not thought more about the graphs described in the comments).

The paper

Harant, Jochen; Rautenbach, Dieter; Recht, Peter; Regen, Friedrich. Packing edge-disjoint cycles in graphs and the cyclomatic number. Discrete Math. 310 (2010), no. 9, 1456--1462

constructs graphs for which the difference between $d$ and the nmaximum number of edge-disjoint cycles is $k$ for any $k \in \mathbb{N}$. If I understand the question correctly this shows you can't have a bound desired without some restrictions on the graphs (I have not thought more about the graphs described in the comments).

Edit: I posted in a rush. This paper and it's references are relevant, but does not answer the question. See comments below.