IELTS is the International English Language Testing System, the most common English Language Testing test in the world.
IDP and British Council conduct IELTS tests but they do not write them. All tests used at the BC and IDP are compiled by Cambridge English Language Assessment who is one of the joint owners of IELTS. This means you will get the same tests if you are at BC or IDP.
No university or college asks you for IELTS-IDP or IELTS- BC band score. What they ask, is just your IELTS band. Their decision is unaffected by any of the bodies through which the test has been taken.
As to scoring, there are two parts of the IELTS exam that are clerically marked – it means that your Answer Sheet is being checked by people who have clear instructions what answers are to be considered correct and incorrect. Reading and Listening modules are such those parts.
Hence, in a nutshell you are free to choose whether you wish to take up IELTS through IDP or British Council, as there is no difference in their exam pattern, marking.
City wise if you see, then you can join whichever best training center available in your city. You can visit the website and reviews and can accordingly make the decision. Medium of training and city of the institute can be any. But you will have to definitely join one to get a proper feedback and guidance, along with self preparation to secure a good band level.