Firstly, tread carefully here even though you are 100% correct. They may look for other reasons to dismiss you once they realise you are right including looking for historical reasons prior to your notice of your intentions.
It’s clear on the Citizens Advice website that they cannot dismiss you early solely on the fact you have given your notice.
If you really don’t want to leave before the 3 months then I would suggest discussing the facts, perhaps try face to face or telephone rather than email, that you want to help the business transition to a replacement employee (hence your giving extra notice period) and that you are 100% committed to the job in the meantime.
You only need to get 3rd parties involved if it becomes clear they are not listening.
Find out how much notice you need to give when resigning and how to negotiate a reduced notice. You can also check what happens to your holiday entitlement.
www.citizensadvice.org.uk
This is a good reason though why I never advise giving too much notice. Give the expected amount and let them negotiate it. But always try to keep it respectable and try to leave on good terms