Word erupted back in November that Microsoft would be migrating businesses to the New Outlook client, and the community is responding with Intune remediation scripts, GPO entries (for hybrid "AD-first" endpoints), and Microsoft itself has added new entries within Intune CSPs targeting 365 App config.
I see most shops that are "registry-first", or still leveraging Group Policies primarily, and so for them I recommend this article for controlling/managing it.
Here is a (new/untested, do NOT run in your production environment without sandbox testing first!) Intune Remediation Script which seems a perfect fit to manage/avoid any surpises on your Endpoints. And the author Martin Bengtsson has a great blog post to accompany it, here.
As for the Outlook Web app, it will also migrate users on the web side of things if you don't set a web mailbox policy to also manage it. More information on that on a MS post here, and it includes tenant PS cmds to set that afforementioned web mailbox policy.
For my employer or any clients, I would recommend slowly rolling this out with the Outlook Web app first, and then the rest in slow groups. Especially if you're running the Semi-Annual Channel (SAC), as you'll have more time to test/manage deployment of the new Outllok.
Good luck!