What position in the university can supervise a PhD? Professor? Lecturer? Associate professor? Post-doc? Assistant professor?

Sweden: The requirement is that the main supervisor  has the degree of Docent. This is however not an employment title, so they'll likely be employed as professor ( = tenured, senior, full professor) or lektor (in this case, it would roughly correspond to associate professor, though there are also lektor without the Docent degree, which then are closer to lecturer). No real equivalent of the assistant professor exists, to the best of my knowledge. I have seen it being compared to quite a few different titles in the Swedish system, but it makes for somewhat of a strenuous translation no matter which of them you go for. So: the closest US-style equivalent titles for people that can supervise a PhD student in Sweden are  '(full) professor' and 'associate professor', respectively.

In the UK, PhD can be supervised by lecturers, senior lecturers, readers, professors, research fellows, post docs, research assistants, and probably some other job titles that I've forgotten.