I checked Minato-kufs information on kindergarten:
FY2024 (started in April 2024) for the public ones, applications were accepted on Nov. 20, 21, & 22 in person, or to be received by 22 if posted by snail mail, and they required that the families are residents in Minato-ku (those gscheduled to move inh were not allowed).
If there had been more applicants, they would have done a lottery, but for FY2024, lottery didnft take place. If the kindergartens are not full, they do accept later applications, but usually by December they want the parents to start getting ready, meaning getting the kids go through health check-up, conducting an interview (with parents and the kid), getting the uniforms ready (if applicable), etc.
The difference between public and private ones seems to be mainly in the fees. (The average figure mentioned in one website is 170,000 yen vs. 350,000 yen.)
Here is a list of private kindergartens in Minato-ku (in English):
https://www.city.minato.tokyo.jp/kyouikushomu/kodomo/gakko/yochien/shi...If you come at the beginning of the year and want get your child enrolled from April right away, asking your employer for house hunting earlier, possibly narrowing down on the area, and asking a private kindergarten if they accommodate your child right away might be easier than with public ones.
When it comes to finding kindergarten with non-Japanese kids or English-speaking teachers, I guess youfd have to ask the individual kindergarten, or ask your employer for their experience.
(Ifm Japanese, and have helped a non-Japanese friend when she wanted to send her child to a child care center, and then onto a public elementary school, even though in a different ward of Tokyo. Unless you send your child to an international one, they expect that you bring a local friend/an interpreter with you, and that you can read all the newsletter and brochures in Japanese that they send out.) I hope someone with experience can contribute with more insight.
Best wishes.