No, like your Birth Certificate, school and other certificates are usually considered to be a “matter of fact”; as long as they were correct at the time they were issued, they will not generally be updated with your new name.
If you need to show your certificates to someone and the name is different, you simply show your Birth Certificate, or some other evidence of your previous name, at the same time.
However, you may want to notify your school or university of your change of name, as you may still be receiving correspondence from them or other associations connected with them.
If you have not already done so, get a deed poll now.