Morning,

I do not find your store in a directory. It's displaying in the public root, no folder.
IF your store is actually in a folder, the redirect is a good idea, but redirecting with .co.uk is tricky because it's technically a sub-domain. Make sure you test the redirect extensively. However, I do not see it in a folder.

I also think you should check the setup your configure files is correct.

Like so
admin/includes/
define('HTTP_SERVER', 'https://www.domain.co.uk');
define('HTTP_SERVER', 'https://www.domain.co.uk');
define('HTTPS_SERVER', 'https://www.domain.co.uk');
Make sure to use the www based on your site's Google indexing.

includes/
define('HTTP_SERVER', 'https://www.domain.co.uk');
define('HTTPS_SERVER', 'https://www.domain.co.uk');
Make sure to use the www based on your site's Google indexing.

admin/includes/
define('ENABLE_SSL_CATALOG', 'true');

includes/
define('ENABLE_SSL', 'true');

admin/includes/
define('DIR_FS_CATALOG', '/home/full/path/to/store/with/trailing/');

includes/
define('DIR_FS_CATALOG', '/home/full/path/to/store/with/trailing/');

admin/includes/
define('DIR_WS_CATALOG', '/');
define('DIR_WS_HTTPS_CATALOG', '/');

includes/
define('DIR_WS_CATALOG', '/');
define('DIR_WS_HTTPS_CATALOG', '/');

~Melanie