I think the way to do it on other phones has been described before, so this is just going over old ground, but I always appreciate step by step instructions all in one place. However, being very noob at this, an obligatory disclaimer is if you try this I have no responsibility if you use this advice and it ruins your brand new phone. Use at your own risk. That said, since you're not unlocking bootloaders or rooting or anything like that, I think flashing sony stock firmware is relatively noob-proof (I'm a living example). I did it on my fresh-out-the-box X compact, so don't know if it will wipe your data - assume it will.
No doubt a professional will be able to cut out half my steps and correct this method, but that's appreciated
As corrected by someone who knows what they're doing (thanks and all credit to AndroPlus and IgorEisberg), you can just download Flashtool, click the Xperifirm button and download the firmware. This lets you skip steps 2-5 below: Flashtool will read the fwinfo.xml file and automatically create the FTF. Then head to step 6 below. Or follow all the steps I listed, which is the long way to do it, but also works
1. Download Xperifirm and flashtool:
2. In Xperifirm find and download the UK customized version of the X compact firmware. This downloads a folder full of files that will need to be converted into a single .ftf file using flashtool in order to flash it to your phone, also with flashtool.
3. Delete the fwinfo.xml file from the folder before you convert into .ftf
4. Open flashtool, go to tools>create> select the folder you downloaded. Select the device (F5321), write in the branding (Customized UK) and version (number with periods). Select all the files on the left and click the arrow to move them to the right.
5. Click create to make the .ftf file in c:/users/YOURUSERNAME/.flashtool/firmwares (or something like that). If it asks you to use the default.fsc select no.
6. Open flashtool. With your phone turned off, hold down volume down and plug in your phone into the PC. Flashtool should say its detected in flashmode. Click the lightning bolt icon, select flashmode, and choose the .ftf file you just made. Hit flash, and again if it asks for default.fsc select no. It might take a while to prepare files for flashing, just leave it. After it starts flashing it will take less than 5 mins.
7. Unplug your new fingerprint enabled phone.
This was all amalgamated from other peoples guides and hard work. For reference and reading first: