I bought an unlocked Verizon phone from the USA - The 4G / LTE doesn't work on 3 mobile. The supported bands are: Bands 3,4,7,13 - Bands 3 are used by 3 and EE with band 7 also used by EE. So, in theory it should work sometimes on LTE.
The AT&T model has LTE bands 2, 4, 5, 7 and 17, so in theory the Verizon model covers two bands compared to just one on the AT&T model.
The phone flips between H on the status bar (no data connection when on H) and H+ (as soon as you demand data, it seems to switch to H+). However, sometimes it just stays on H with no data connection.
I'm on Windows 10 insider preview now, build 10.0.10586.36 and the data speed seems a bit improved over WP8.1 (I used speed test and have achieved almost 8Mbps down 4Mbps up compared to max 2-3 Mbps down and less than 1Mbps down, not scientific but I've seen an improvement).
I'll also test the EE network as it works on two bands (3,7) compared to just band 3 on Three mobile.
As other posters have pointed out, there's no MMS support, although apparently you can go into developer mode, download an app and adjust a registry key to reveal the MMS settings (works for Verizon Nokias but nothing said about HTC).
Also, speed seems to improve if you tick off "Use this APN for LTE and replace the one from my mobile operator". However, still no 4G speeds.
If I let the phone select its own APN instead of manually putting it in, then it won't work well on data for some reason.
The person I bought it from said he called Verizon and and they advised it would work on 4G with either a Verizon sim or a UK build of the Windows Phone software (of course there isn't any official UK build for the HTC One M8 as it wasn't sold in this market).
I'm hoping later builds will help with the connection but won't hold my breath. I'll try the fix for MMS which is over in the Nokia Icon forum and see if that helps.
If anyone has had success with 4G / faster data on the HTC One M8 Windows in the UK then please let us know how you did it!