The title says the query is related to CBSE, hence I take CBSE first.
The CBSE registration for the XII board exams starts off in the XI standard itself. This is carried out around October of the XI year. If the subject change is done before the registration process is completed, the school authorities themselves can do it. You need to approach the principal for this.
If the registration has been sent, then you need to contact the CBSE board with a detailed letter requesting the change. Subject change is easy rather than change of stream (difficult and not advisable at the CBSE).
If the CBSE rejects this request, then some students choose the option to switch over to the state board for the 12th.
Once you pass Class XI, then change is not allwoed
In individual/rare cases, the CBSE chairman has powers to authorize a student to take a new subject provided it is done before 30th of August.
So, I suggest that you and parents have a discussion with the principal first and then plan it.
For the state board, please contact the principal first who can guide you and this is possible if planned before August (when the student is in XI)