Some Muslims do this practice of reciting the Salaat an-Nabi or "Darood Sharif" as it is called in the Indian subcontinent. Is this halaal according to the Book of Allah (the Qur'an) and the Sunnah please?
Why shouldn't it be accepted if Allah () says:
Indeed, Allah confers blessing upon the Prophet, and His angels [ask Him to do so]. O you who have believed, ask [ Allah to confer] blessing upon him and ask [ Allah to grant him] peace. (33:56)
This verse doesn't make any exception or restriction on when or where one should praise the Prophet (). How to send prayers and blessings is covered in almost all tafssirs of this verse (see for example here in qtafssir)