Grrrr. So I did another live chat with HTC to try and find out more about which models they sell in the UK. The Amazon listing has a model number and I was hoping HTC would tell me that's the model number for the LTE variant so I could go back to Amazon to say they'd mis-sold me the wrong version. This time though I got a different story; the HTC rep told me my handset is definitely LTE capable and what options you see for networking are tied to what the SIM card is capable of. His assertion was that Three had given me a 3G only SIM card and getting that sorted would fix it.
However, I swapped SIM card with my colleague who's getting a 4G signal from Three and it made no difference on my handset; I still only got the same 3G network options and, of course, only got a 3G signal. It also occurs to me that I checked with another colleague this morning who's got an HTC One but only on a 3G contract with O2 and he's already got the LTE network options (although he only ever gets a 3G signal of course).
So, yeah, this second call to HTC was a load of nonsense. Excellent. Back for a third time I go...