Update: The plot thickens. We currently have exactly 2 answers, opposite each other.
I couldn't find the answer on fandom, but this has been asked on quora and reddit. I wasn't able to find anything sourced.
https://www.reddit.com/r/harrypotter/comments/4c5syq/is_hogwarts_the_only_school_in_the_uk/
https://www.quora.com/Is-Hogwarts-the-only-school-of-Witchcraft-and-Wizardry-in-Britain
https://www.reddit.com/r/harrypotter/comments/3kkpyk/are_there_other_wizarding_schools_in_the_uk/
FYI some context:
I'm asking this as like a lemma for my other question: Why wasn't the Daily Prophet reporting the attacks? If Hogwarts was the only magical school in the UK, then I'd think the daily prophet would be interested a series of blood purist magical attacks in hogwarts.
Also I'm wondering which schools invite(/accept) any given wizard/witch/sorcerer/sorceress/warlock. I could've sworn there was a question like this on reddit, quora or stackexchange, but I can't seem to find right now. (This isn't what I'm wondering. This is sort of.) If there are other magical schools in the UK, then I'd think they'd invite(/accept) Harry too. But maybe they just stayed out of Hogwarts' territory since Harry's parents attended Hogwarts. (Then I could generalise this to all magical schools on Earth like every single school sends a letter to every single 11 year old witch/wizard, regardless of region/territory and so for Harry they all stayed out of territory.) But anyway this is another question post. (btw, this is referred in (3) below.)
Also, I'm wondering about transfers. No need for another question to see if this is a thing this time because of these: reddit, se, fandom, se again. What I'm wondering here (ie/aka the part that is for another question post) is the practicality of transferring. Hell if I would let any witch/wizard kid or younger sibling of mine continue in Hogwarts after there's been this sorcerer's/philosopher's stone attack, basilisk attack, werewolf/dementor attack or whatever even if it would be a hassle to go to Ilvermorny. Plus 2 of the biggest (UK, but in this scenario I'm a UK parent. Bloody hell!) dark wizards actually attended this school, so probably I would forbid my kid/younger sibling from attending this school in favour of other schools (assuming any letters were given i guess...see (2) above. or maybe the letters aren't automatic. maybe you have to actually apply unlike with harry to hogwarts). Preferably, Hogwarts has competitors in the UK. Or maybe the lack of safety in Hogwarts IMPLIES it hasn't any competitors in the UK. But anyway, again, this is another question post.