Here's my experience, over two days last week, at Tortosa to get an NIE...

Arrive outside (good free car parking) by 06h15. It's not the actual police station, but offices on Passeig de Ribera.

An informal list is available (at least, it was on both days I was there), and make sure your name is on it. Look to the right of the door, on a shelf, under a stone. Someone queueing may be there to help you. If cold, it's fine to sit in your car, until 07h50 at least.

My experience was: if you are 33rd on the list (first day), you probably won't get in. If 16th, you probably will. Spanish nationals are sorted at 8 am into one queue, and everyone else (EU and non-EU) into another. How many in the 'other' queue will get through the door will be determined, I think, by the number of Spanish applicants on the day and perhaps by the number of appointments.

Come prepared with your form. You can type in the details online and then save it as a pdf and print it out. Make sure you have details of the property you are buying available. Remember your passport!

You will be given a printed number. Watch the number display like a hawk - they won't wait unless you go to the appropriate desk very promptly (format is something like 0417MESA10, meaning number 0417 should go to desk 10), and you will miss your turn.

Once seen and processed, you will have to take a form to the bank to pay for the process - 16 euros, thereabouts. Caixa will not do it unless you have a bank account with them. Santander will. There are two branches within 10 minutes' walk.

Head back to the office by two. On production of the receipted form, you will get another form with your NIE handwritten on it. This is not the final document, but it can be used for legal purposes before the notary - at least before the notary I went to.

You have to go back three days later for your formal document. Between 09h00 and 12h00. No queueing. No printed number needed.

My impression was that the beginning of the week was busier than the latter end, but...

... if you want the NIE the same week, I suggest you go for the first time on a Monday or Tuesday, and you'll get it on the Thursday or Friday.

You can make an appointment for a day sometime ahead. When I went, they were offering them about 10 days hence.

Crazy system - but it does work.