A friends TDI300 is smelling of burnt oil about a minute after startup.

It is an automatic so part electronic.

Any ideas please?

its burning oil !

Oil leak onto any part of th exhaust manifolds?

Slipping belts can smell a bit oily (ish)
My alternator belt stinks a bit when it's a tad slack but don't squeal.

Oh the smell is coming out of the exhaust

Well oil is getting into the combustion chamber and burning, the fact it only makes the smell for the first wee while could just mean that after it's warmed up you are just not aware of it as the combustion is more complete.

How many miles does the engine have on it, what grade of engine oil is in it. It could be the rings are worn and oil is coming up from the crankcase - just a little which is why its notability lessens after a short while.

It could be an indication that it's your valve stem seals are worn and oil is leaking down - not too bad to fix and can be done in situation.

I would start by taking the inlet manifold pipe off and see if the inlet pipe looks very oily if I was fairly certain the engine wasn't just worn, if the inlet pipe is oily then think turbo or crankcase breather. If the inlet pipe is OK then think more internal such as the valve stem seals. You would need to pop the inlet manifold off and have a look into the ports to see if it's coming down the valve guides.

Valve stem seals leaking overnight ?