Related information:
Get an Apple Music family subscription - Apple Support
How to switch to a different subscription plan - Apple Support
I don't know and Apple does not specifically say. However, I strongly suspect the change would take place immediately and you would then receive a pro-rated refund for the unused part of your individual subscription.
If you absolutely need to know, ask Apple:
Refer to this page for Apple Support features ➔ Contact - Official Apple Support
Select from the presented options until you find a solution for your issue, or see if there is a chat or phone call contact method offered lower on the page (you sometimes have to narrow down the options multiple times before this is shown). If you do not see your issue, keep experimenting with any series of selections until you reach one that offers a chat session or a telephone call and get the representative to redirect you.
This page has telephone numbers for contacting Apple for support and service ➔ Contact Apple for support and service - Apple Support
By the way, in future it would help if we knew to what you are subscribed. Apple has many different family plans.