For the purpose of this post I have assumed that your retaining the existing walls and slab. Planning permission will not be required but should the works be more extensive then you could need planning permission.
Building Control will need to be notified and the proposals of your works issued to them. I would recommend using an "Approved Inspector" over your local authority as in my experience they are quicker to respond and alot more pragmatic. Estimated cost - £800.
I assume that you will be installing a cavity membrane system on the walls and floor (most basements require this) to keep out the moisture. This action activates the Party Wall Act.
Fixing the cavity membrane system into the party walls will require you to serve a Party Wall notice on your neighbours on both sides. Should you neighbour "dissent" to the works then they can appoint their own surveyor or you share a surveyor - but your on the hook to pay the surveyors fees (estimated £1,500 per neighbour).
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