My team is remote so we use slack and other apps all the time, but getting customer service or other depts to use a simple chat is difficult. Implementing something like Asana or Basecamp is impossible.
How did you implement those tools in your company?
You have to remember that tooling isn't valuable for its own sake. A chainsaw is much more powerful than a knife, but it's the wrong tool when you're sitting down to dinner.
Getting new tooling adopted in the enterprise is about sitting down with your users, understanding what their goals are, understanding where their current pain points are, and showing them how the tooling will help them achieve their goals and solve their pain points, oftentimes requiring close, high-touch support and adaptation to get that done.
If you can't sell the tooling on its own merits to people in your company, you have no business mandating its use.
That assumes that a) such dept has the capacity to understand the advantage of using such tool and b) other depts have time to go and try to sell them that.
Users always resist change.
A little while ago I wrote some docs for a project, a simple markdown readme basically, but the boss decided that all documentation has to be in sharepoint, so I converted it to a word doc and threw it in sharepoint. My takeaway lesson was not to not document anything though because I hate using sharepoint and Word for.
A top down command like "use x or get fired" isn't particularly helpful, making the right thing to do the easiest thing to do generally is though.
"What is the status of issue X?"
"What is the IssueID?"
"I didn't create one"
"Create an new entry in the tracker and we can all track progress of issue X"
Repeat as many times as necessary.
edit: I also try to create a trigger or other automatic action in the tool. For example, our issue tracker updates the release notes when items are modified or marked as completed. If you can get people to use the triggered tool it will create an incentive.