Although it's been never directly addressed, it is quite plausible
Firstly, the only Wizarding schools that we know of are:
Hogwarts School of Witchcraft and Wizardry (Scotland)
Beauxbatons Academy of Magic (France)
Castelobruxo (Central Brazil)
Durmstrang Institute (Northern Europe)
Ilvermorny (Eastern North America)
Mahoutokoro School of Magic (Japan)
Uagadou School of Magic (Uganda)
So, there are many countries and regions that don’t have their own Wizarding schools. It is highly plausible that wizarding kids could have been accepted at Hogwarts because, well, wizards have been known to live in other regions where there are no Wizarding schools.
About the fees structures, no one pays to go to Hogwarts; the only fees are those for textbooks and other necessary materials. In fact there is even a funding for those who cannot afford to buy those necessities.
Moreover, Draco Malfoy bragged about how he wished his father would send him to Durmstrang where there was no admittance for muggle-borns. So, it is the parents who decide where the kid should go.
So answering the first part of your question:
It is not necessary to be born in the U.K to be accepted at Hogwarts. One could be born in America and if the parent/guardian submit application at Hogwarts (and it is accepted), they can.
NOTE: Hogwarts has also been known to have permanent exchange students as mentioned in the answer here. So internationals admitted in Hogwarts (in a different way though) - Possible.
Speaking about What if they "first exhibits signs of magic" while on vacation abroad?
From the movie Harry Potter and the Sorcerer's Stone, it is mentioned by Hagrid that Harry's name was added to the Hogwarts' list since the day he was born.
HAGRID: A great Muggle like you is going to stop him?
HAGRID: (to Harry) Non-magic folk. (to Vernon) This boy's had his name down since he were born. He's going to the finest school of witchcraft and wizardry.
So, your school would send you an Owl accepting your admission at Hogwarts no matter where you are (if of course you have been enrolled by your parents before coming of age).