Hey Nandini

Both British Council and IDP conducts IELTS examination under the authorization of Cambridge and so the it does not differ much between British Council and IDP. Both have very similar test pattern and format and you can choose any one of them, you can consider the Exam center and timing, which suits you more. Most of it depends on your performance, how much you score in IELTS exam and does not matter on the agency (British Council/IDP).

For further doubts, ask us. All the best