She was pretty prim and proper in The Thin Man that same year (and young looking; only 22 when she made the Tarzan sequel). Quite nubile here for a nice girl from Ireland. Tarzan and His Mate
was just barely pre-Code, but one of the films that certainly led to strict enforcement of the Production Code; Breen and Hays raised quite a fuss. Not only is she scampering around in barely there "jungle" apparel, sleeping in the nude (and clearly enjoying premarital bedtime frolics with Tarzan), but after all this sexual freedom she's credited as "Jane Parker" (vice "Mrs. Tarzan").
They shot three versions of the four minute nude swimming ballet featuring Olympic swimmer Josephine McKim
doubling for Ms. O'Sullivan, one with the costume, one topless, one nude except for a skin-colored "apron" (merkin?) in place. The alternate cuts were apparently available for regional theaters to splice in to comply with community standards/laws. The fully nude scene was restored along with a quick male nudity shot when Ted Turner bought the MGM library and an uncut positive print was discovered. O'Sullivan went on to have seven (!) children with John Farrow, including actress Mia Farrow and Prudence (for whom The Beatles wrote a song, "Dear Prudence"). Her grandson Ronan is a Pulitzer-winning journalist, known for his "#MeToo" inspiring articles in The New Yorker and other pieces.
Your DVD is possibly missing 12 seconds or so of lions being deliberately tripped to simulate being shot, reportedly removed by British censors in 2002ish. Not sure if it's in the latest versions.
, IMDb says the restored version is 116m, Leonard Maltin's TMC review says the restored version is 105m. How long is yours?
Interesting pic on Wikipedia of the film's engagement at Nanking, China.