Yes it is possible and is not at all uncommon.
All you need to do is to select your new school and talk to them. They will want to know what parts of the training you have already completed and what exams you have passed and failed. The new CGI will use this information to advise you on how much of the course you need to do.
If you have already used up a lot of exam attempts, but have had lots of failures the new CGI will probably advise you to terminate your current exam series and start a new series. The CAA will usually agree to this if you are changing schools. If however you have used only one or two attempts and had a reasonable degree of cuccess, then it would be better to continue with your current series. Although the new CGI will provide advice, you must decide what you want to do.
Once you have signed up with the new school the CGI will inform the CAA of the fact that you have changed schools. The CGI or the CAA will then ask the old school for a copy of your training records to be sent to the new school. You will need to sign this request. The CGI will then use this copy to confirm that what you have told him about your previous training is true.
Then it is simply a matter of completing the training and passing the exams!!!