Thanks for all the suggestions received on this.
The shipping is now complete, to London, and here's the report. (If anyone is interested in maybe one of the "Most Unlikely traveller's bike shipments" ever).
By the way, I'm a "bystander with big interest" in this project.
Enquiries with Ethiopia Airlines (suggested by Achim) led to a Khartoum freight company: AGS SUDAN.
Part of the French "AGS Movers" international freight company.
They had never carried a private vehicle before but had carried vehicles for UN and NGOs.
They didn't know about the Carnet system but understood once it was explained.
They have a depot at Heathrow.
Everything was done through the Sudan manager, Frederic Blanchon, based in Khartoum, who's French and speaks English plus lots of other languages.
He was very helpful throughout and provided an initial estimate of price in USD.
The Yamaha Serow was collected 'as is' by AGS SUDAN, with the Carnet and UK logbook. No dismantling or palletising or crating, but the battery had to be removed and left behind. Tools, a bicycle and some other personal effects were included.
As far as we know, the carrier was Lufthansa.
With the bike in AGS's depot, Frederic Blanchon gave a firm price in USD and GB pounds (the exchange rate seemed reasonable and the price was a bit less than the estimate).
This included delivery within 100km radius of Heathrow.
About 10 days later it all arrived at Heathrow. The Heathrow office of AGS Movers needed some advice on clearing through customs which was straightforward after Frederic Blanchon had advised them.
A date was booked for the whole lot to be delivered to Eastbourne and it arrived on time, delivered by Brittania Removers, the first we knew of their involvement in this (obviously sub-contracted by AGS Movers).
The bike was exactly as it was collected in Khartoum, on its wheels, no dismantling. Plus stamped Carnet, log book and bicycle, tools and other stuff.
So it was pretty successful. But took a lot of email exchanges with AGS Sudan to organise the details and get a final price.
The payment was done over the phone, by UK bank debit card, in GB pounds, to AGS Movers' HQ in France.
The price - if you want to know - well, send me a PM. I'm the 'bystander with BIG interest'!
Never before has the term "Job done" been so meaningful.
For info, AGS Sudan website:
AGS Frasers Sudan | AGS Worldwide Movers
(Which reminds me - job not quite done - maybe I'll try to find out where our tree is...!)