Change name in the passport

I recently got naturalised as British citizen and obtained my passport.

Now I would like to change my name. For that I have got an unenrolled deed poll and a statutory declaration.

When I make an application to change the name, should I make it under new name or old name and then specify the new name in the application?

You apply under your new name, for a new paspport, wih all the necessary documentation. Your existing passport doesn't get updated.
Please refer to the HMPO Policy regarding name change: https://assets.publishing.service.gov.u ... ations.pdf