What a basement impact assessment actually is
The phrase you typed is borough-specific, and that is the first thing to understand. Some London councils, such as Camden, have a formal, audited document literally called a Basement Impact Assessment. Kensington and Chelsea does not use that name at all. In the Royal Borough, the same job is done by a Construction Method Statement, submitted under the council's basements policy.
A basement impact assessment is a technical report, submitted with a planning application, that sets out how a proposed basement will affect the things around it. It covers the structural stability of your own building and the neighbouring ones, the ground itself, water in all its forms (groundwater, surface water, drainage and flooding), and nearby trees. Because it spans structure, ground, water and trees, it usually draws on more than one specialist, not the architect alone.
The exact name for it varies by council, which is where the confusion starts. In Camden, Basement Impact Assessment is a defined, audited document with that precise title. Kensington and Chelsea does not use the phrase at all. In the Royal Borough, the same content sits inside a Construction Method Statement, usually shortened to CMS, which is the engineer's report describing how the basement will be built and what effect it will have.
The CMS is required under Local Plan Policy CD11, formerly Policy CL7, and the more detailed Basements Supplementary Planning Document, or SPD, which is the council's adopted guidance that expands on the policy. The SPD was adopted in April 2016 and remains the operative detail. So when someone in Kensington and Chelsea says basement impact assessment, the document they actually need is a CMS.