University Admissions Advice

Hello, first time poster (not that that's super relevant, but whatever)
For context, I'm a Scottish Sixth Year student taking Advanced Higher English and Modern Studies, and Higher Politics and Media. I already have Higher English, Modern Studies, History, Human Biology, and Philosophy; and National 5 English, Maths, Applications of Maths, Modern Studies, History, Chemistry and Drama. All of the subjects mentioned above that I have completed, I achieved A's in. Additionally, though I'm a British national I studied in Turkey most of my life so only started in Scottish education at fourth year (I am consequently bilingual)
So, I've got several thoughts about uni that I need to figure out fast:
As options in Scotland, I have pretty much narrowed it down to St Andrews' İnternational Relations course - I have the grades, and I think with a good personal statement I should be good enough to at least get an offer of some kind. I'm also considering whatever Politics course Edinburgh offers, or maybe somewhere closer to home as insurance like Robert Gordons University (this is admittedly a slim chance though, I intend to leave Aberdeen).
The difficult part comes now: Above those, I've also been considering Oxford and Cambridge, but I've got several issues here. First, like an idiot I decided against taking even Higher Maths - so even though I do consider myself pretty good at Maths (the Turkish Maths curriculum is ahead of the British one by a lot) I know that one of my two preferred choices at Oxford (PPE) will be pretty difficult to convince without that qualification, considering they recommend Advanced Higher Maths. On the other hand is History and Politics, the other course I'm considering, which recommends Advanced Higher History (not so bad given that I at least have a Higher), but also requires two pieces of written History work, which I don't really feel that I have beyond standard Higher History essays. Another annoyance is the admissions exams for the two are different, with PPE wanting TSA and HP wanting HAT. There's also the personal inconvenience that I'd have to delay a holiday to take the HAT, but that's not so important in the grand scheme of things.
For Cambridge, the option would be the course that's name I can't remember right jow but is something along the lines of Humanities, Sociology and Politics and also requires written work to be submitted, which again I don't feel confident that I have at an acceptable level.
So, any advice?

I'm not sure what your concern is with the history written work for Oxford. You're supposed to submit one piece of written work, and it should be completed as part of your regular school studies. Your school has to certify this. You can't just write an essay on a random topic of your own volition and upload that.

I'm aware that it has to be school work certified by my school (although I mixed up how many I needed to submit, thanks) but my worry is that since I haven't taken Advanced Higher History and only Higher History, that my written work would just be... Lame? Maybe that's a stupid concern but still.

I'm not too familiar with the Scottish system, but if they require the advanced higher, they'll say so. As for the quality of your written work, I don't think that should impact it very much. There's nothing stopping you from writing some revision essays now, getting them marked by your teacher, and selecting the best one. I would recommend focusing on whatever area of the course you find most interesting, because they'll likely ask about your written work in the interview, and it's really obvious to tutors if you're bored and uninterested.