Q: Apple ID country

To change the country on your account (whilst you are here in the UK you can only use the UK store) you are likely to have to enter UK payment details (you should be able to remove them after entering them) : Change your iTunes Store country or region - Apple Support

Why won't it work, why can't you log into the account ? You are following the instructions on that link, logging into the account, selecting the country/region section and selecting UK, and then entering your UK billing details ? If you can't change the country on that screen then do any of those linked to scenarios apply ?

What happens if you click on the 'click here' link that partly shows at the top right of your last image, does that not let you change the country ?