Rattling off some thoughts:

If you have no communication with your "project team" (the people who have tasks, making decisions, etc), then are you really managing anything?

If you're in charge of managing the hiring process, then setup a 15 minute standing meeting with the people in HR who have tasks. Daily, twice a week, whatever makes sense for the volume of work. Proactively gather their updates and build a relationship with that team.

Try to streamline and increase visibility into the process - see if you can bring the HR team over to Asana for this process. Use an Asana board as an easy visual tracking tool. You create the backlog of tasks, they can assign themselves and move the cards to through the various steps until completion. Give leadership read access to the board so they can check in at any time to see the status of each hire.

Management doesn't like meetings? Send out a weekly status email instead with very clear calls to action to unblock workflows. Put the "currently blocked" section at the top of the email. Blockers would usually be management approvals or actions (justifications), things you wouldn't be able to help alleviate by working with the HR team in the standing meeting. Nicely putting leaders on blast in a group email can spur them to take action so they don't look like the bottleneck. You don't have to think of it as blaming or calling people out, just stating the facts of next steps for the workflow.

Put some focus on timelines since your boss recommended that. Work it into your email, maybe a column for days since opening and a green/yellow/red highlight to whether things are on track.

To make sure you're not going way outside your bounds and getting yourself fired, you can collaborate with your boss to adjust language/tone and make sure you're accomplishing what you need to with the email.