Wikipedia switched from UseModWiki to its successor software in 2002. The year is the canonical answer, although the change happened in stages rather than on one single day.
The first major step came on January 25, 2002, when the English Wikipedia moved from UseModWiki, known retrospectively as Phase I, to a PHP-based system called Phase II. Magnus Manske developed that software specifically for Wikipedia, using a MySQL database and adding features such as namespaces, watchlists, file uploads, and automatic signatures.
Phase II soon encountered performance problems as Wikipedia’s readership and editing activity increased. Lee Daniel Crocker rewrote the software into Phase III, which went live in July 2002. This version was designed to scale better while retaining much of the earlier interface. The name MediaWiki was suggested in 2003 and gradually adopted from August of that year, so some detailed histories distinguish the 2002 technical migration from the later naming change.
That distinction explains why dates such as 2003 can appear in discussions of the software’s history, but 2002 is the accepted answer for Wikipedia’s move away from UseModWiki.