This link ended up being really fantastic, the commands are the same for moving from 12 (which I was on) to 15, and I presume it'll be the same for future upgrades
$ pg_lsclusters
$ sudo pg_dropcluster 14 main --stop
$ sudo pg_upgradecluster 13 main
$ sudo pg_dropcluster 13 main
$ sudo apt-get purge postgresql-13 postgresql-client-13