The service you experience with Prime definitely depends on where in the UK you live, or rather where the item is going.

I live in a city centre, within 50 miles of an Amazon distribution centre, and it often offers me same day delivery. Like, order at 3am, items arrive at 8pm.

But if I change it to my parents' address (more remote but still in civilisation), it takes them 1-2 days to deliver.