Hello and thank you for your question. I will be very pleased to assist you. I'm a practicing lawyer in England with over 10 years experience.
Can you kindly confirm that you have had sole possession - i.e. no one else has had nor could have had access to the basement other than with your permission through your flat please?
Thanks. On the basis that the basement is not already demised to a third party under an existing lease (in which case the rules are a little different) you have been using the basement exclusively and no-one else has had access for a period of 12 years or more you may have a case for adversely possessing the basement. If you are successful in claiming adverse possession of the basement on the above grounds - you will likely need a solicitor to assist you with a adverse possession claim - the basement will be added to your demise and will be subject to the same terms as the lease - as provided for by Smirke v Lyndale Developments Ltd.
The first step is to confirm that the basement is not registered to another leaseholder. You will then need to amass evidence to support exclusive possession for 12 years or more. This can be in the form of witness statements and / or other evidence such as photos or documents showing you have had occupation. A solicitor can and will in practice help you with amassing evidence.
If you can show the required evidence as above then the basement can be added to your lease as above and a registration application to alter your title deeds at the Land Registry.
Is there anything above I can clarify for you any further?