Your SIM is taught to rename networks, think IP addresses and DNS names - they are essentially the same thing, just 2 ways of representing the same address. Sorta.
Really all cell towers broadcast a number (in T-Mo's case - 30) and the SIM knows it can connect to 30 and when it does it knows it should say it's connected to "T-Mobile". But the network share between Orange and T-Mo presents a problem - the other network (34) calls itself Orange, but the phone can now connect to it too, so what should they call that? Well they settled on "T-Mobile Orange" (Or vice versa if your primary network was Orange) to avoid people panicing, because in reality the public would probably cry if they saw the wrong network first.
So back to current day, they need to program the SIMs to call both networks EE and be done with it (and also say either network is allowed - currently there is a priority system) but SIM updates can fail, a million other things can go wrong, masts may still broadcast the wrong thing, a whole bunch of stuff can hold it up.
In reality - you should do nothing, it's largely a symbolic thing to customers and has more of a meaning for the maintainers of the network. The service you recieve will not be affected by this happening/not happening largely, the only real change is the one duffman might have spotted.
