The issues I've had (in the UK, indeed) has not been with the delivery time itself (which is often stated, correctly, as "next day" or "2-3 days") but with the time to process my order, some times taking several days. Most severely, two months, as they had neglected to account for the time it takes to customs process a container from China. But the items did arrive one week (plus two months) after the order was placed.

But yes, a lot of small vendors do ship very quickly. They do still generally require me to create an account and unsubscribe from their newsletter, regardless of unticking the box during signup.