If its small and comes in via Royal Snail AND you have to pay customs duty then you will probably get a card through your door giving you the good news. If you pay online delivery can take another 3 or 4 business days. The quickest way is to go into the sorting office and pay and collect.
If it is USPS Expedited and comes in via Parcel Farce then as Dave said it can take 4 or 5 days for them to actually get the card and pay the duty, then arrange for delivery.
If it is Fedex or another big courier then you will get the goods delivered on schedule and an invoice for duty will follow.
You have more chance of getting it in without duty if the sender uses normal USPS Airmail going through the Post Office rather than expedited going through Parcel Force. If they use Fedx then you are going to pay duty
The longest we have waited is about 6 weeks and the shortest time has been next day via Fedex International Priority.
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When I die, I want to go peacefully like my Grandad did, in his sleep -- not screaming, like the passengers in his car