A word for first name and middle name/s, but not last name

First name:

the name that was given to you when you were born and that comes before your family name:

• It can be rude to call people by their first name if they are much older or more important than you.

(Cambridge Dictionary)

Full name

The Full Names of Everyone in the Royal Family
King George VI, born Albert Frederick Arthur George (1895-1952)
Queen Elizabeth II, born Elizabeth Alexandra Mary (1926-)

Now some might opine that full name suggests also a person's family/last name (surname) and it's a possibility, especially since middle names are becoming less frequent. Therefore, the clearest and simplest phrase is to say: first and middle names.

Try personal names

: a name (as the praenomen or the forename) by which an individual is intimately known or designated and which may be displaced or supplemented by a surname, a cognomen, or a royal name

Merriam Webster

You might say forenames for first and middle names

forename - A person's first or ‘Christian’ name (OED)

In my lifetime, the two most common words used to describe names other than the family name are given names and Christian names. The latter has been steadily falling out of favor. Indeed, a google ngram view of the two shows exactly that phenomenon: n-gram of christian name, given name. Although, it should be noted, in common usage both labels are applied to mean the first name, or commonly used name (should it be other than the first) only. There is no single word, in English, that will automatically be understood by the recipient of a communication (verbal or written) to mean "all the names a person has not including their family name".

(Edit: for an interesting comparison, add forename to that n-gram view.)