When {{#statement:Pxxx}} is used on a wiki page to generate an external link, it always consumes the "formatter URL" of the Pxxx property. (The same applies to Lua functions like mw.wikibase.formatValue.)
When this "formatter URL" statement is updated, the pages have to be re-parsed (purged) to show the correct link.
Use Wikibase usage tracking to deliver these updates (e.g., aspect C.P1630).
Note that the option formatterUrlProperty is only available to the repository, which makes this somewhat non-trivial.
See T112081: [Story] purge cached renderings of IDs when the formatter URL changes for a similar story on the repository.