This is getting silly now. There is also some incorrect information being given.
First off Mantababe, you need to speak to the dealership. Until then, anything is speculation and is therefore little use to you. We can't answer it all, neither should we have to. The dealer is being paid by you and Fiat, so they need to do their job.
Secondly, Fiat do not issue documents to enable registration of a vehicle. The vehicle moves through various statuses on Fiats system, from when it's ordered to when it's delivered. Once it reaches a certain stage the dealer is able to link it with a customers details and a registration number via AFRL (Automated First Registration & Licensing), which is a computer link to DVLA. At this point they are able to print a document which contains the vehicle and customer details and the registration number. The customer is to check all the details are correct, sign and date the document and the dealer then confirms the registration on AFRL, which sends the information to DVLA for them to print the V5C registration certificate/log book and post it to the customer.
Registration numbers cannot be picked at random by the customer as Goudrons as correctly pointed out. You might be lucky if your initials match, for example my initals are LM, so if I bought a car from London I might end up with LM64 or LM15, but otherwise it's likely the reg number will mean nothing as they are allocated at random through AFRL to the dealer. For example a dealer may have their area prefix and age identifier (in Oxford that is OA-OZ, then currently 64) followed by a selection of random letters usually following a pattern, ie AAA, AAB, AAC. They might not have AAZ, that might go to another dealer, and its highly unlikely they'll have anything that means something to most people. That's what private plates are for.
Therefore the issue here isn't so much being able to choose a registration number, as the dealer is effectively powerless there. The problem is they appear to have registered a car on a 64 plate, when it might not turn up until such time that it could've been a 15 plate. This you will only be able to check with the dealer. If it turns up this week as the salesman said, then no problem. However, if it turns up after 15 Feb, then you could potentially have asked if they'd hang on to register it as a 15 plate for March 1st. They might've said no, especially as you've had a hefty discount. However if they haven't provided you with an order form with any terms and conditions, they're on dodgy ground in saying what is or isn't allowed.
A 2015 64 plate Panda is likely to be worth a few hundred quid less than an equivalent 2015 15 plate in 3 years time, so you could possibly ask for compensation in terms of money or accessories/fuel. Around £250 is reasonable, any more is a bonus. If they turn down any request you could choose to cancel your order, but again I don't know where you stand with this as they've not provided you with an order form. You would need to speak to Citizens Advice or a Solicitor if you decided to go down this route.
Another problem for the dealer is that if they have registered the car without you seeing the AFRL documents with your details on, they're in breach of their AFRL agreement. The documents with your details on are what form the registration document sent from DVLA, and DVLA take a dim view of errors on registration documents caused by the dealer not checking details with the customer. If they found out your car had been registered without you checking the details first, they could suspend the dealers AFRL license, meaning they would legally not be allowed to register any more cars - serious stuff. Whether DVLA would resort to this for one breach I don't know, but if a number of errors were made because of it, they probably wouldn't be best pleased, and they do audit dealers who have AFRL licenses to ensure they're complying with the rules.
I know all this as it's been my job to register cars on AFRL for a number of companies over the years including a Fiat dealer, and I've also worked for DVLA. Therefore apart from allowing for any changes to legislation in the past 9 months since I stopped doing it for a living, I will be correct.