Is this what the URls look like normally? (not from UK so I would not know)
Why not just https://ee.co.uk.billing.info/jan-02 ? I mean if someone does not notice it is a domain name and not path would they really notice where the info part is?
In addition to the sibling comments, there's also the problem of having an actual slash in your example, which may cause the mark to notice it and therefore notice the "missing" slash after the 'co.uk' part.
Because ee.co.uk.billing.info would get taken down pretty quickly. Using this method the domain only has to work for one day before they move on to the next one.
Why not just https://ee.co.uk.billing.info/jan-02 ? I mean if someone does not notice it is a domain name and not path would they really notice where the info part is?