Well the UK doesn't have a constitution in the sense of the USA, so technically there aren't any rights there in that sense.
But, yes, some countries enforce their laws on companies there that deal with anyone. If you're not in the USA or Canada and have a Facebook account, Facebook is bound by Irish Data Protection (i.e. privacy) law. You, as a non-Irish citizen, or resident, talk to the Irish Data Protection Commissioner and get them to enforce your rights. And Austrian person did just that: http://europe-v-facebook.org/EN/en.html
But, yes, some countries enforce their laws on companies there that deal with anyone. If you're not in the USA or Canada and have a Facebook account, Facebook is bound by Irish Data Protection (i.e. privacy) law. You, as a non-Irish citizen, or resident, talk to the Irish Data Protection Commissioner and get them to enforce your rights. And Austrian person did just that: http://europe-v-facebook.org/EN/en.html
Additionally in the UK, you are entitled to some forms of free medical care from the NHS, even if you're a non-UK resident just visiting: http://www.nhs.uk/NHSEngland/AboutNHSservices/uk-visitors/Pa...
So no, not all systems work like the US system.