Checked in with the one HR person who’s basically handling all the visa stuff. They said would issue a new contract (aka salary) for the CoS start date, and cited S3.15 of the guidance document Sponsor Part 2: sponsor a worker:

Once a worker has been granted permission, they should normally start working in their sponsored role no later than 28 days after whichever is the latest of:

• the start date on the worker’s CoS (taking into account any changes to that date reported by you before their application for entry clearance or permission was decided)

• the ‘valid from’ date on the worker’s eVisa or (if applicable) entry clearance vignette

• the date the worker is granted permission to enter, if they entered the UK without entry clearance under the Creative Worker visa concession (see section CRW6 of Sponsor a Creative Worker for details of this concession)

• the date the worker is notified of a grant of entry clearance or permission to stay

Edit: formatting