I have got it all sorted now. It's not that difficult, a short summary of what needs to be done, perhaps you will find it helpful:
1. HMRC - The first thing to do is to let HMRC know you have imported the car permanently into the UK. You shouldn't have to pay any tax as there is no way the car is new.
2. MOT - If there is nothing wrong with your fiat, the only thing that will make it fail the MOT is the fog and reverse lights (they will have to be swapped) and the head lamps (stickers or black tape will suffice - there is no need for new head lamps). There is also no need for the speedo to be in miles. The MOT certificate will have the VIN number on it.
3. Once it's MOT-ed, send your application to DVLA to register it - the form you need is V55/5. Include the MOT certificate, the original log book (you will not get it back), cheque for tax £75 and registration fee of £55. You don't need type approval (as the car is over 10 years old), and contrary to what it says on the form, you don't need to insure the car before you register it. Insuring the car based on vin is very expensive. I advise you to wait for the registration number from DVLA and then insure it.
You will get your V5C in post within a few weeks. The form says 4-6 weeks, but I got mine after 1,5 weeks.
4. Now you can insure it. It may take time until insurance companies download your registration number from DVLA which may make it more difficult for you to insure. The systems which the insurance companies use on their websites do not recognise any 126 made after 1992 so you will have to speak to them. My first quote based on VIN was £525. With the registration number I paid £270.
5. Enjoy driving. It's a lovely car.
I hope this will help somebody.