If you want Fiat dealership parts online then you will be directed to a Fiat dealers website. I don't know of any other way to get Fiat branded parts. However if you want OE spec parts then factors may be able to help but most UK factors will not spend time, and money putting parts online with a limited market. They will all probably be able to get the parts but will not show them online.
From a dealership point of view, although parts are available overnight, the manufacturers give a very limited discount for this service to encourage the dealers to stock parts themselves.
A dealer will only stock parts that sell often.
Therefore, if you go into a dealership of any brand and want an odd part immediately you will have to pay the price. If you can think ahead and order the part and negotiate with the parts man that you will have a bit of discount if he puts it on his stock order. He will get a far bigger discount this way and may be willing to pass some on.
It may be worthwhile considering getting your parts from the same dealer all the time. If you are a regular customer, and with a Fiat you are going to be one, then you may get a discount each time you visit. If you tart yourself all over the country then do not expect any help, or more importantly, discount.
Cheers
D