As far as I am aware, you will need to obtain a temporary export licence for your car from the Export Licensing department of the Arts Council: see website at:
According to the details on page 24 onwards of the document on the second link, you will need a licence for your car, even for a temporary exportation as it is more than 75 years old. This document contains phone numbers and contacts of people who should be able to help you & advise in more detail.
In addition to an export licence, you will need to clear your car through Customs whenever it leaves the EU, and then enters or leaves a non EU country. The easiest and quickest way (if not the cheapest) of doing this is to obtain a document called an ATA Carnet, which is like a one-time passport for the car, and prevents you from having to pay duty or taxes at each importation. You can find out more about this, and purchase a Carnet from your nearest Chamber of Commerce.
You might also wish to check out whether your car will be subject to any equivalent of road tax in any of the countries you pass through. normally A valid UK road tax will exempt you from any overseas road tax requirements, but I guess your car is probably not covered by normal UK road tax rules. This is not an area which I have experienced, but you may be able to find out more by contacting a collectors' asociation for classic or vintage cars, as many of them seem to organise rallies & international trips.
I hope that this helps.