What is the current state of async and task management solutions (collective.celery, plone.app.async) for Plone 5.2?
Not sure about that.
I believe that collective.celery might work with little or no work if you pick this branch https://github.com/collective/collective.celery/tree/py3plone52
plone.app.async is definetely a no go for Python 3 (async will not be importable, see https://github.com/plone/Products.CMFPlone/issues/2721)
collective.taskqueue will support TxZServer (Twisted ZServer) once its ready
I'll continue on TxZServer once Plone supports the latest Zope 4.x or Zope 5.x. (The return of WebDAV into Zope core makes polishing of my ZServer fork into publishable TxZServer possible. Meanwhile I'm collecting some production experience from the current ZServer version.)