@Christian Müller the "up next" view in the "Home" section which is available on iOS kind of does this.
You can also now use the Calendar power up on mobile, which shows you cards in a calendar view on a single board.
You can also approximate a due date sort of all cards assigned to you across all boards using the search feature on mobile.
For example you could search:
@me due:overdue
to see all overdue cards assigned to you
@me due:day
to see all cards assigned to you due in the next day. You can use due:week, due:month too.
The ordering will not be in date order but if you're looking at only a handful of cards that won't really matter.
However a due date sort of all cards assigned to you across all boards the same as the web app is something you'd need to submit as a feature request:
https://help.trello.com/article/724-submitting-feature-requests-for-trello
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