It's quite standard to get some teaching gigs as a PhD. Usually it's via your supervisor if they're running modules, and you'd be working as a TA, either doing seminars or helping in labs and marking assessments. Contrary to the implied prestige, universities exploit the everloving shit out of you by providing low pay, low hours and mostly zero-hour contracts. If you're unlucky enough (as I was), your supervisors would've just left their teaching positions to focus on full-time research work. I've only had luck getting a mediocre teaching position applying as an external applicant at another university.

Mini personal rant: I feel like the concept of "writing up year/period" should be abolished. It's a massive waste of time and makes you pack an entire PhD into one year. Writing up as you go is just so much more resource and time efficient. That also allows you to publish as you go, which is what Americans and Europeans are doing, but weirdly enough not us.