Did a little research, checking IMDB and BoxOfficeMojo, and here’s what I’ve found:
*The movie premiered Christmas Day 1993, in the middle of a seven month hiatus in Season 2. While the series never really addresses the events of the movie, and it was made while the series was going, you could place it there, if you really wanted. A lot of other episodes don’t get addressed after the fact, either.
*Joker just–came back. No details on how he got away from Andrea or anything like that. The end of the movie also shows that she made it out alive from the same explosion, so the how is probably best left a mystery, like Bats disappearing so easily or whatever specific object Spider-Man’s web is anchored to for each swing.
*The closest we get to seeing Joker fall into chemicals was during The New Batman Adventures (or New Gotham Adventures, or Gotham Knights, as the final season was sometimes called), when the animation shifted to more closely resemble Superman’s series at the time. In the episode “Beware the Creeper”, there was a reenactment of the incident for Jack Ryder’s report on the Clown Prince of Crime, but Joker’s personal experience is never depicted in the series.
Why, you may ask? Because the series came out shortly after “Batman Returns”, and was kind of a loose adaptation of the two Tim Burton films. This is why Penguin has webbed fingers Catwoman is a blonde, and in one episode, “Dreams in Darkness”, the lead doctor at Arkham mentions Jack Napier, Joker’s identity from the first Burton film, implying that in the cartoon, Mr. J had a clear backstory and everything. [Important Note This is a one-time mention, and is never brought up again. In fact, in “Beware the Creeper”, Ryder mentions that Joker was a member of the Valestra mob, as in “Mask of the Phantasm”, but also says that his identity was “erased”, as if implying he somehow destroyed all traces of his identity before becoming Joker.]
Tangentially related: In the cartoon series, “The Batman”, a stylized flashback shows that Joker falling into chemicals, and it get’s mentioned as what made him who he is in a later episode, but I don’t think they ever mentioned Batman being in/directly responsible.