I did a search earlier today to see if I could find an answer to the question "Will an unlocked US GSM iPhone 5S work on a UK 4G/LTE network?" I did not find any actual results of testing, merely speculation, so I thought I'd look at the specifications of the phones and the networks to see if I could come to any conclusion. Since posting, I found this thread, which indicates success in using a US iPhone 5S both in the UK on O2 and in Norway on 4G networks. Based on this empirical data, I plan to buy a 3 PAYG SIM in the UK since their network has two frequencies compared to O2, has good coverage in London and seems to have an attractive prepaid package for data and voice.
RichmondDan: According to https://discussions.apple.com/index.jspa:
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When I posted, based on the lack of actual data about the ability of an A1532 iPhone 5S to work on an 4G network in the UK and this description, I thought that sharing what I researched might qualify as a tip. In fact, I specifically unmarked this thread as a question.
varjak paw: Yes I am aware of http://www.apple.com/iphone/LTE/. In fact I referenced it in my original message. I'm not sure what more there is besides the radio hardware, an unlocked handset and a correctly provisioned SIM, especially since AT&T has an 4G/LTE roaming agreement with EE. The LTE roadming capability indicates that the A1532 will work on the EE 4G network (but only on band 3). Again, per my original message, I realize that the A1532 does not cover the bands used by all UK 4G networks, however, I just care to find one UK 4G network that it will work with so that I can buy a PAYG SIM for it for my next trip.
Peter GIllespie: Model A1457 for European countries covers LTE bands 1, 2, 3, 5, 7, 8, 20. Model A1532 for US and Canada GSM networks covers LTE bands 1, 2, 3, 4, 5, 8, 13, 17, 19, 20, 25. The US and Canada do not use LTE band 7 whereas Europe does. Presumably there is a technical or commerical reason that Apple manufactures a version specifically for the US and Canada, and another for Europe. I am not disputing that. I am merely trying to ascertain whether I can use LTE in the UK on my US iPhone using a PAYG SIM.