So, the code to do this in the Dialog component is pretty trivial:

// create a new ref to stash the focus value when dialog opens
const previouslyFocused = ref<Element|null>( null );

When the Dialog's onDialogOpen method is called, add the following lines:

previouslyFocused.value = document.activeElement;

And when the Dialog's onDialogClose method is called, restore focus via the following (the extensive conditional is here to keep typescript happy)

if (
    previouslyFocused.value &&
    'focus' in previouslyFocused.value &&
    typeof previouslyFocused.value.focus === 'function'
) {
    previouslyFocused.value.focus();
    previouslyFocused.value = null;
}

This all works fine. However, in cases where a Codex Button component is the trigger that opens the dialog (which is probably the most common use-case), the button styles start to break: you don't see a blue outline as expected, and the button remains stuck in the active styling state:

{F56298138}

This is a pre-existing problem in Codex, but fixing this Dialog issue has made it obvious. It looks like the setActive() code that was added to the button a while back does not handle this situation very well.

Change #1052820 had a related patch set uploaded (by Eric Gardner; author: Eric Gardner):

[design/codex@main] Dialog: restore focus to previously-focused element on close

https://gerrit.wikimedia.org/r/1052820

LGTM! Thank you!

Change #1052820 merged by jenkins-bot:

[design/codex@main] Dialog: restore focus to previously-focused element on close

https://gerrit.wikimedia.org/r/1052820

Change #1056999 had a related patch set uploaded (by LWatson; author: LWatson):

[mediawiki/core@master] Update Codex from v1.9.0 to v1.10.0

https://gerrit.wikimedia.org/r/1056999

Test wiki created on Patch demo by LWatson-WMF using patch(es) linked to this task:
https://patchdemo.wmcloud.org/wikis/1c9a4708c9/wiki/Main Page