In the original, the that clause, that former TV Apprentice contestant Katie Hopkins was speaking at a Brunel University debate, is the complement of the adjective angry: it tells us why the students were angry.

That's why angry follows students instead of preceding it: adjectives which are themselves followed by a complement or modifier have to be placed after their "heads", the nouns they modify, so that the complement or modifier doesn't fall between the adjective and its head.

Your rewrite changes these semantic and syntactic relationships.

In the first place, you have made Ms. Hopkins' presence the object of the protest instead of the cause of the anger. In this case that's probably justifiable—it's a reasonable inference that the students were protesting what made them angry. But it may not be the case; the protest may have been directed at something different, such as the administration's intruding an unwanted speaker into a student-run event, or the administration persistent refusal to consult them in scheduling speakers.

In any case, if you do make this change you have to adjust the that clause to the sort of clause that the noun protest takes, typically a preposition phrase headed by against. And when you move things around you have to tidy up the mess you leave behind: you have to move angry, which no longer has a complement, back in front of students; and you have to delete in protest at the end when you move it to the beginning:

In protest against former TV Apprentice Katie Hopkins' speaking at a Brunel University debate, angry students have walked out.

Almost there. If you want to put "In protest" at the beginning, you need to follow it with "at"+gerund.

Also, you seem to have put in protest both at the beginning and at the end of your proposed change (not intentionally, I presume).

A correct way to form that word order would be:

In protest at former TV Apprentice Katie Hopkins speaking at a Brunel University debate, angry students have walked out.