Congratulations on your wedding.
So you do have a gKosekih in Japan - you might want to call the city hall in Japan (or the embassy of Japan in the UK) to make sure what process is needed.
I know that after the baby is born outside Japan, youfd need to report the birth within 3 months of the birth. Also that (if the baby obtains another nationality by (1) being born in another country or (2) being born to a parent of another nationality) you need to make a report to reserve the Japanese nationality also within 3 months, otherwise by the fact that the baby has obtained another nationality he/she might lose the Japanese nationality. (Ð¯ÛÌÍ¯)https://www.mofa.go.jp/mofaj/toko/todoke/koseki/index.html
Ideally your marriage should be registered first in your gKosekih then the birth (avoids complication as to whether you were married at the time of childbirth, but you already are by the UK law, so that should be no problem once your marriage is registered).
As far as gKosekih goes, your marriage to your husband will be noted in the gremarksh column of your gKosekih as gMs. Hanako Suzuki got married to Mr. John Smith (British national) on Nov. XX 2021 according to the UK law.h (He does not get entered into the main text, as he is not Japanese, but your marriage to him will be duly noted.)
Your baby will be entered into the main text (I believe), as a Japanese national. Name difference should not matter - your koseki will have the fatherfs name mentioned in it.
Just by the way: One idea might be to (when you renew your passport) ask the Japanese authority to add your spousefs family name in parenthesis after your maiden name on your passport. Your gkosekih name (= official name) remains the same, but just for convenience (to show who you are married to, when you travel outside Japan) there is a way to have it so that it appears as (for example):
- Hanako Suzuki (Smith).
I have mine done like this. (I am a Japanese woman, my spouse is an EU national, we got married in the UK and reported our marriage later to Japan, I kept my maiden name.)
Best wishes for you and your family.