Screen time password was forgotten but there’s no option for a “forgot password” at the bottom even though phone is updated fully?? Help??

Regarding your data, If…

1. You have an iCloud or iTunes backup, I am sorry there is no use of a backup because if you restore from your backup your Screen Time Passcode will be back. 
2. You have synced your data to iCloud, then what you will get back is what you have synced to iCloud and uploaded to iCloud Drive; like the information that are synced and stored in iCloud are, Contacts, Calendars, Bookmarks, Mail, Notes, Voice Memos, shared photos, iCloud Photos*, Health data, call history, and files you store in iCloud Drive. (*-If you had turned on iCloud Photos, your content is automatically stored in iCloud.)
3. You have imported your Photos & Videos to Mac/PC then they are safely residing there
4. You have uploaded some of it Cloud Drives using Files app (iCloud Drive, Google Drive, OneDrive, DropBox, Flickr, box, AmazonDrive, Adobe Creative Cloud etc)
5. You have not imported nor you have synced nor uploaded your data to cloud drives prior to forgetting the Screen Time passcode then I am sorry, all your data is lost.

If you are unable to reset Screen Time passcode from the this-> link then, you will need a computer. If you don't have one, borrow from a friend.

Forgot? Click below to reset…

Click->If you forgot the passcode for your iPhone, iPad, or iPod touch, or your ...

You may not have enabled Apple ID for Screen Time or may have updated the device after setting the Screen Time Passcode.