It is the usual Amazon issue that customers expect the item on the date set by Amazon when they order. Even if that is the day after ordering and they are miles away from anywhere.

Firstly, "Where's my stuff?" - as you say, is the email choice as set by Amazon. I think that is a terrible way to show, but Amazon chose that wording, not the Buyer. Amazon 'should' encourage Buyers to be polite, but they promote rudeness, asking for free items, etc.

Amazon will have shown delivery on/by 29 March. Not estimated, etc, as really they 'should' do, as unless everything is sent by Special Delivery, then Amazon know couriers do not guarantee a delivery date.

The reason why the Buyer is rightly unhappy, is due totally to Amazon, as they will have sent 'their' customer an automated email with a new delivery date of 2 April. The problem is that is a made up date, as neither you, nor the courier provided a new delivery date. Amazon made up that new delivery date. When you reply to the Buyer make it very clear that Amazon made up the new delivery date, and that it was not you, nor the courier that provided that to them, but you will try and resolve the issue, track, etc for them.

I'd apologise that the item has not arrived, and once received that they may raise a return request, then once received back with you, they may have a full refund, as long as new (suggest they do not open the package, so it is then as new).

Track the parcel, if not tracking at all, then you may find there is an issue to resolve. Lkely though, just a delay and tracking ok. Explain to the Buyer which courier used and the tracking number, and to track directly on the courier website, as that often shows updates, delivery times, and the Buyer can alter to deliver to a neighbour, etc.

I assume you are a new Seller, as this issue is long outstanding, Buyers often get this email from Amazon, and likely never will be resolved by Amazon.

Amazon 'should' but won't: (a) show delivery dates as either estimated ... date, or aim to delivery by ... date. and (b) Amazon should email the Buyer saying sorry not delivered yet, track on the courier website and if an issue contact the Seller for further details, and not just make up a new delivery date, which is very unhelpful.

Things to consider:

Is it best to use Buy Shipping if possible, that way partly protected against negative reviwes about late delivery.

Which courier did you use, are they any good?

Did you ship on the correct service, eg DPD Next Day if needing it there the day after.

Did you use an automated shipping tracking service, so Amazon supply the delivery date, as less issues if the item is late if they decided the date and not you just choosing 1-2 day delivery, etc.

If using Royal Mail Tracked 24, DPD etc, you may also find that they will deliver on a Sunday, so track the parcel, it may arrive today, when they want the item, tell them this if that is the case.

If a Premium delivery, 1-day or 2-day, this will count against you, maybe not offer a 1-day service and only 2-day but send on a 1-day for example. If an extra charge refund that amount of course.

Explain to the customer you shipped on time, as proof by the courier tracking. You can add an attachment, with screenshot to prove the courier had the parcel on time and any issue as to why the parcel was delayed.

Be nice to the Buyer, that way likely no bad feedback. It is not their fault, the courier was late and Amazon showed the delivery date as if it were guaranteed.