Hey, Pete!
Maybe it's not the most ideal, but does this help?
https://trello.com/search?q=sort:due%20is:open
The only drawback is it doesn't show the due dates in the list; there seems to be no way around that.
The other drawback is that cards that were due in the past but not archived would show (if that applies to you), but there are more filters listed that can help you refine from there (you can see them by clicking the icon with the "i" in it next to the search bar).
https://support.atlassian.com/trello/docs/searching-for-cards-all-boards/
There's also the "Up Next" section in the "Home" (on the left side) tab on plain old trello.com (once you sign in)
However, I'm not sure how the cards are sorted there.
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