The networks actually transmit an ID code, the first three digits are the country code for the UK (234) and the next two are the network code:
234-33 = Orange
234-30 = T-Mobile (formerly one2one)
234-15 = Vodafone
234-10 = O2 (previously cellnet / bt cellnet)
(Many more listed on Wikipedia).
However how the HANDSET interprets this depends on the handset software, and the SIM card. Some older handsets ignore the SIM card and display the name they are programmed with - others let the SIM override the name, but show an old name for a while then replace with a newer name.
EE has a couple of times sent SIM Update messages that reprogramme the SIM to show the 'EE' name for both 234-33 and 234-30 transmissions. But again, not all handsets support this reprogramming of the SIM.
One reason EE went with keeping T-mobile masts was because the majority were now merged into the MBNL 3G network share agreement with Three, and had hence had big upgrades already installed. 100mbps backhaul (or more) and provided fast 3G services. Lots of orange masts were on 2mbps backhaul or worse and so 3G from these sites was very slow.
Mobile Broadband Network Ltd (MBNL) is originally a 3G Radio Access Network share between EE and Three, and also a mast site sharing contract, and backhaul sharing contract. This meant one equipment cabinet for 3G, one set of antennas on the mast, for the two separate networks. For 4G both Three and EE use their own equipment at the mast site, but share the site and backhaul.
Cornerstone Telecommunications Infrastructure Ltd (CTIL) is different and is essentially just a site sharing scheme, where Vodafone and O2 are installing their own backhaul network connectivity and their own equipment cabinets, but one set of engineers may install both sets. Then sharing the tower with the transmission panels on.
In areas where CTIL has upgraded both networks generally have good 2G,3G, and 4G - but in areas without upgrades the 3G network on Vodafone is often a joke, and on O2 quite slow. (like legacy orange).
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