I bought an LG G2 D802 (32Gb). This model is described as an International variant that works in any geography. So bought one from ebay given the price was so low. I even ordered a new sim card from 3 as I needed a micro sim. The phone arrived and was easily configured to work in the UK. All was good. It was only when I headed in to London the next day that I realised I was not getting a 4G (LTE) signal.
Turns out there are multiple variants of the D802 and I had the D80220A-AME-XX variant. Basically, it seems the hardware for the D802 device is all the same but the AME seems to be a Middle East variant of the ROM. When you try and do a software update from the phone or using LG's software on a PC, it'll only ever install the middle east variant. From what I understand, it's the IMEI that LG use to determine the software. No matter what I did I couldn't get a 4G connection unless I did this hack which doesn't survive a reboot.
After some surfing on XDA-Developers it looked like I needed to flash the EUR (europe) ROM variant. I was in two minds about returning the device and buying from the UK. But the hassle of doing a return back to Hong Kong on ebay wasn't enticing. So I flashed the EUR variant. It's a bit of a hassle as you have to trick the LG software in to doing this. I followed the instructions here and am now running D80220b-EUR-XX. Immediately I'm able to connnect to 4G with no issues at all. In all other respects the phone behaves the same.
I used "LG G2 D802 V20D (Europe Open) KitKat Firmware"
After reading some more it seems my problem might be an issue with 3 (Three) in that the carriers have to explicitly allow phone models to connect to 4G and the D80220A-AME-XX is not on their white-list. Other owners on other carriers don't seem to have the problem. And I assume many users don't even realise there are different software ROMs for the D802. Why LG even bothered to do multiple versions escapes me.
Anyhow - I hope ths helps those with a similar issue.