Question about using different names?

I am in the UK and I have a different name on my Passport than my school documents. I use my middle name and last name on pretty much everything else, but I use my actual First name in the bank and on my passport.

Is this legal? If not, should I call up my previous schools and ask them to change my name? Also, I've applied to university using my middle name and last name. will this be a problem?

Here is an example of my situation.

The name on my passport is Hewlett Jon Parkard. The name I use as my first name fist name in school documents is Jon and the last name is Parkard