I went next door and connected my router to their outside antenna - and the results were slightly worse than I was getting in my attic. I only got 2 bars and the speed was slightly down at about 15 Mb/s.
Three and EE may share the same mast, but their antenna panels can be at different heights which can give different effects - this may explain why using the antenna made things worse. Also the antenna may be slightly directional and pointing at a mast that only provides EE services (maybe with Voda or O2 as well).
Three has less spectrum capacity for 4G signals, which means if a lot of people in your area take up the service then they have less capacity to share. This is why you're getting slower speeds. EE has a lot of spectrum capacity (as does Vodafone) in comparison - which is why they can achieve 50 megabit/s and faster speeds in a lot of the country.
plusnet unlimited fibre
80/20 since 2 Jun 14 - Sync as of 7th Aug 16: 55,355/10,291 kbps with G.INP
17 years of UK broadband since 1999 ntl:cable modem trial -Router: Asus RT-AC68U with HG612 - BQM