Hello and thanks for using Just Answer.
I am Al and am happy to assist you with your enquiry.
You would need to execute a Deed of Variation so that the garden is included in the basement flat Lease. This is a relatively straight forward document for your Solicitor to preare but you would need to provide him with a suitable Plan showing the extent of the garden which is to be included. The Plan will need to be Land Registry compliant- ie to scale and have a North pointer.
I hope this helps and clarifies matters for you.