How do you handle a late delivery well?

Tell them before they ask. A message that arrives ahead of the customer noticing converts a failure into evidence that you are paying attention; the same message after they chase you does not.

Why the answer is what it is

Then give a specific new date rather than an apology, and something small if it slips twice. Most complaints are about silence rather than about the delay, which is why the fix costs almost nothing.

What this looks like in practice

Most support questions are the same five, and answering them before they are asked is cheaper than answering them after.

Unclear delivery timing causes more abandoned carts than price.

Delivery is what a customer remembers, and it is the part most stores control least.

Where a mobile app comes into it

An app makes the after-sale part easier: order status, reordering and support in the place the customer already has open.