My contract with EE was due to end next month. I decided, that as I was quite happy with my phone and did not need to upgrade, to change to a SIM only contract. EE wanted £10ish for their cheapest option, whereas BT was only £5. So I decided to cancel my contract and request an unlock code from EE. EE were very helpful and it would be no problem, but why was I leaving ?. I explained that BT was cheaper, I now have a £5/month SIM only contact with EE giving me more minutes that the BT option. The chap said there was no point in EE losing a customer to BT when the two companies were now one.
They also changed me over a month before the end of my contract, saving me another £20.
So if you are looking to change to SIM only, quote BT's price, you may get what you want without the bother of changing.