OTHER USERS EXPERIENCING THIS PROBLEM. PLEASE READ THIS POST!!!!
The problem is a bug with the Outlook.com android app. There is an apparently easy interim fix that has been identified by other users with this issue in other help forums here. Do not uninstall/reinstall at least until you try this. You risk losing all of our unsent/outbox-stuck messages if you uninstall.
The problem seems to be that certain messages - apparently if you are offline (i.e. not wi-fi connected, etc.) (not *sure* of this, but other users have identified this as the source of the problem - go to the outbox and never get sent, even when back online.
Then, all susbsequent messages sent queue up behind this message and also get stuck in the outbox, even if you are online.
(In my case, the "problem" messages also had attachments on both devices, but that may just be coincidence.)
But as other users have stumbled upon, if you go in and delete just the oldest message from the outbox, it seems to then fix the problem and all the emails get immediately sent. (Although on one device I had to delete the two oldest messages before this worked.)
But it seems VERY clear to me that this an app bug, and has nothing to do with configurations, operating system versions, etc. (See my next post, answering the moderator's query about by set-up, etc.)
Good luck all!
OTHER USERS!!!!: PLEASE TRY THIS WORKAROUND FIRST!!!! DO NOT UNINSTALL/RESINSTALL UNTIL YOU HAVE AT LEAST TRIED THIS. YOU MAY BE ABLE TO AVOID LOSING ALL YOUR UNSENT OUTBOX MESSAGES, WHICH WILL ALMOST CERTAINLY HAPPEN IF YOU SIMPLY UNINSTALL.