What it'll mean is there isn't a formal union recognition agreement with the school. This is quite normal for academies- and often what it boils down to is a formal union recognition agreement usually asks for reps to have a measly one hour a week to focus on being a rep during school time, which the school doesn't want to pay for. With large academies, unions that have formal recognition agreements are allowed to engage in collective bargaining for all staff, whereas unrecognised unions aren't. Generally unions like NEU and NAS would be recognised, whereas e.g. GMB may represent a small number of support staff, but may not be recognised as having bargaining rights with the academy.
I've been a rep in a school without a formal union recognition agreement. The head still met with me, and the academy CEO and head of HR acknowledged my role in certain circumstances. They even allowed me time to go on a training course to be a rep! The lack of a formal agreement was literally because the agreement asked for me to get a small amount of paid time to focus on my rep duties and they couldn't make that work with the timetable, and it meant I wasn't allowed to negotiate on members' behalves over pay- but this was a bit moot as they did pay the national pay scales!
If, e.g. they want to make redundancies, or to discipline a staff member, then there are legal rights around union representation. If the union votes to strike in a legal way, then all the laws around striking, picket lines etc could still apply.
In some schools, I imagine this could be used to refuse a regular meeting with reps, but I think this is a really bad idea for the school and could lead to bad feeling.
Schools cannot legally bar staff from union membership, nor can they stop union meetings from happening, although they can stop these from happening on school property.
I hope this helps a bit!