Full disclosure: I work at Replicon, and use its products. Time tracking software is our core business and we offer both SaaS and customer-hosted implementations. The Project & Billing edition of our Web TimeSheet has significant tools for setting up projects and tracking tasks, and you can get pretty granular about that, but the product is not intended to be a full-featured project management app like MS Project (although it’s easy to share data with any app that has an open API). I think it would be worth a few minutes of your time to read more about it. Then if it seems viable to you, sign up for a free trial (which is SaaS-based, for 14 days). It includes sample data, and you can enter you own if you like. You get full support during the trial, including 1:1 with a product expert who can answer your questions. We’ve been around since 1996—1.5 million users in 70 countries.
Some open-source ERP software like project-open seem to have all the needed capabilities, but implementing it throughout the company seems like a huge headache…