It is doubtful that Fiat UK will have much influence on local dealers. They would outsource it to their local favourite, possibly based on price (to them), speed of service, or just convenience of the nearest.
If you mean Fiat breakdown, that will probably depend on the breakdown agent attending, who will choose based on similar criteria as above.
If you have a puncture, you are free to choose. Any local tyre shop should be able to view and advise on repairability of any tyre damage. Although personally, I'd avoid Kwik Fit.