Actually, it's a bit less complicated than that.
MBNL was the joint venture between the UK arms of T-Mobile and Hutchinson 3UK (aka 3).
MBNL formed a further joint venture with Orange that they called Everything Everywhere.
At this point, anyone on a T-mobile, 3 or Orange contract weren't actually on any of those, they were legally on an Everything Everywhere contract. However, the same brands were used to prevent shocked customers ('cos we can't handle change, can we :x ).
Whatever network you are on, you are using a combination of all three at the moment.
Everything Everywhere (presumably because they all got tired of having to type so many letters when they were spelling their name) have now re-branded as EE.
They are also in the process of upgrading their network in certain cities to 4G which is very fast data download. You can only access this if you have a 4G compatible phone and 4G doesn't carry voice calls, so when you get your snazzy new phone and want to speak to someone, your phone will use the existing 3G network.
Other networks - Vodafone and O2 - are joining the party next year at some point.
It's also worth remembering that the 4G network runs on a higher frequency so you won't have to suffer the slow speeds/poor coverage that T-mobile and Orange currently suffer due to their 3G networks running on lower frequencies. And as 4G is only available in certain large cities, coverage should be very good on 3G within those areas anyway.
All of the above is open to correction if anyone is a technical engineer.
For what its worth I have a Mac, a Windows laptop and a smartphone, all of which are trained to buy a round in any pub I am in; although the laptop sometimes forgets the order when it gets to the bar and my smartphone spends too long showing off to pretty barmaids to be of any real use.