Hi
I started my training at Liverpool John Lennon in a Tomahawk, and changed over to Wickenby and converted to a Cessna 152.
There's not much you have to do really. Just let your current flying school know you're moving of course, and introduce yourself to a new flying school, and let them know how far you've got with your training. You will have to get your new instructor to ask to post your training records from your old flying school, as you're not allowed to have them yourself for some reason.
Make sure you've got your licence application form with all the ground exams you may have done signed.
Also, make sure your log book is totally up to date and signed/stamped otherwise it can cause problems when you send away for your licence.
Once I moved to Wickenby, I just got introduced to my new instructor, spent a couple of hours getting used to the different plane, the new airfield and also the new area I was flying in. Then just continued from where I left off with my training.
Hope this helps.