Short version: it’ll come down to the ticketing airline, not the metal.
Even though you’re all on the exact same SQ operated flight, Jewel early check-in usually follows who issued the ticket. Since yours is ticketed by SQ, you’re fine. The Air New Zealand ticket is where it gets dicey, because Jewel staff tend to stick pretty strictly to the operating rules of the marketing carrier.
In practice, this means there’s a decent chance you’ll be allowed to check in early and your family won’t. Sometimes agents will make an exception if it’s clearly the same flight and same PNR group, but since you’re on separate tickets, I wouldn’t count on it.
Worst case scenario is you check your bags early, your family has to hang onto theirs and do a normal airport check-in later. Best case, a chill agent lets everyone through anyway.
Changi staff are generally great, but codeshare edge cases are where flexibility disappears fast. I’d plan assuming only the SQ-ticketed passenger is guaranteed Jewel early check-in, and treat anything else as a bonus.
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