My work has a number of bureaucracy. For a request, a number of approvals must be made, and after approval, my request will go to different teams for implementation. All those approvals and implementation by each team is sequential, meaning that if there is a bottleneck at some point, it will delay the subsequent process. Even for one request, it can take more than one months to do it. After all teams are done with their implementation, I will start doing my part of work.
I'm looking for advice on how to improve this situation to be more efficient. I've been asking the person that is handling the task like a few days after they are assigned (i.e. if 2nd team is assigned, then I will ask regarding the progress a few days after 1st team has completed the work). I'm not sure whether they will find me naggy or something, but I really need to get the job done before the deadlines, and I don't want my work to miss the deadline just because of these kind of bureaucracies.
When being asked, some of them they just say that they are busy with their work and can only get back to me like the week after. But here is the problem, if 1st team is busy on the first week and hence can complete on the 2nd week, and the 2nd team is busy on the second week and hence can complete on the 3rd week, then it's very difficult for me to finish it as soon as possible.