I'd like to know if it's possible to use NFC on an HTC One X to pay for things in the UK and if so, what app do I use?
Google wallet doesn't seem to be supported here in the UK.
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A more generic answer, not specific to the UK alone:
As Ross pointed out, there is not "one specific app to rule them all". That payment stuff is very much vendor-specific – whith "vendor" meaning the company dealing with the payments. Unless several shops/companies join up for a "standard", it might very well be you'd need a separate app for each shop even.
So this question cannot be answered by a "this-is-it" app, but rather would become a pretty long list (Ross' answer already indicates such), which would be subject to (almost daily) changes whenever a new "vendor" joins or another gives up.
To give you an example: Here in Germany I know of two grocery chains using NFC payments, each using its own app – though both belong to the same "mother corporation". If not even those can figure a "joint approach", I very much doubt there will be a "global approach" any time soon.
NFC payment support is still pretty hit and miss in the UK. "Cash on Tap" and "Quick Tap" seem to be just pre-pay NFC cards you can load into your phones NFC chip, and not many phones are supported (your HTC One X is not).
If you have a BarclayCard, you can get a "Barclaycard PayTag NFC" sticker for your phone which seems like a better solution since it will work on any phone and it is actually linked to your card. It also claims to work on any contactless till. There might be other banks doing this too.
Worth noting that contactless card support is still hit and miss too - I've found some big stores don't advertise it but it works (eg Tesco) but others claim to have it and it actually doesn't work (eg Co-op).