The LBS doesn't have anything in stock to replace the crank on this bike :
What is this type of crank and does anyone still supply them?
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It's hard to tell from the picture, but it looks like a cottered crank. I don't think anybody makes them anymore, but not all hope is lost. You can likely replace the entire crankset and bottom bracket assuming the bike has a bottom bracket which is still available, which is somewhat likely. It may cost more than you are willing to spend on the old bike though.
These crancks are still very common on Dutch vintage bicycles. You can order them from many Dutch websites starting at about 6 euro's. All Dutch bicycle shops can order such a cranck at every wholesaler. Many even have such a cranck on stock. You may want to look for a Dutch website which is willing send it to the UK before you replace everything.
This cranck is called a 'cranck met spie' (cranck with spie) in Dutch. If you find one, make sure the lenght and thread are the same as yours.
Edit: there you go: link. Still check whether the length and thread are the same though. And buy a new wedge from the same website, don't use the old one.
When mounting such a cranck, don't try to get the wedge further in the cranck by screwing the nut tighter. Grease everything, and use a hammer to get the wedge in the cranck.
(I don't know if wedge is the correct word. Is gusset or forelock better?).