Job Management / Tracking

We are an accounting firm and we are looking for a software / database system that will help us manage client deadlines and email clients if certain actions are outstanding. These emails would be sent automatically until the stage is marked complete and the mail would vary depending on what stage the job is at.

I've attached an excel spreadsheet of a simplified version of what we would be looking for.

I used to use something similar within a Lotus Notes Database but this is going back 20 years or so.

I'm looking for someone to choose a good software / tool and then design the database / tracking elements for me.

Thanks