As an update for anyone else struggling:
After yet another long call with sky and speaking to 3 more departments, my case has been assigned to a manual eSIM team. Apparently they're back office so they can't connect me but they'll be in touch with me within 2-3 hours. Importantly, they can apparently call you on a non sky number for anyone stuck on that part of the process.
Not sure why I couldn't be transferred to this team beforehand and I don't have any great faith they'll be able to help given the amount of time I've spent trying to resolve this so far and the amount of people on here having similar issues. I was meant to be leaving for a trip in an hour and have had to delay this to try and get it sorted.
If you are in the same boat, I thought it might help to know which team to ask your case to be assigned to.
