Just to update this.

My fibre broadband connection was smoothly switched over this morning. Didn't have to make any changes to my connection on the ONT or reconfigure my router.

I waited until I had confirmation from the 2Degrees of my connection date before providing notice to One NZ, which I did via their online cancellation form. Their form has a "Reason for Cancellation" dropdown box and has "transferring to new provider" as an option.

I received a pro-rated invoice from One NZ last week to cover internet charges until yesterday.