What is the best upper bound on the running time of the graph minor algorithm?
My answer was based on the explanation in paper of Adler et al for disjoint paths problem. That running time is not exactly what is in the Robertson and Seymour paper but it is the improved one by Kawarabayashi and Wollan. It’s been a long time ago and I am not sure if this is restricted to specific graph class. You may have a look at the corresponding reference.

https://arxiv.org/pdf/1310.2378.pdf