Wow... Thank you very much for the answers and the discussion! Just to clarify my situation:
1. EASA CPL with ME/IR, issued in 2013.
2. No FI rating yet.
3. Never held UK licence before.
4. The employer is sponsoring FI training in Britain. Up to my knowledge, the ATO that I am going to do the training at has only UK CAA approval to provide training. I assume that it should be FI rating into a British licence. Hence, the reason of getting a British licence for myself. I will investigate their approvals (is there some database to check what kind of certificates ATO has?), as many useful comments have been made in this discussion. If it is possible to fill in two sets of papers (one for EASA and one for UK CAA) in the same exam, why not.