Oxford's own live Wikipedia infobox "type" field (confirmed via raw wikitext) lists only "Collegiate research university" and "Ancient university", with no public/private designation, unlike University of Cambridge's infobox which explicitly states "Public collegiate research university". The lead section and a full-article search of Oxford's own page for public-funding language were also checked live and found no citable public/private statement.
What would resolve this A verbatim statement, from University of Oxford's own Wikipedia article or an equivalent primary institutional source, explicitly classifying it as a public or private university, or a registry decision to source ownership-type for UK ancient universities from a different, explicitly named source class.
OpenUniversity of Oxford (Wikipedia)