Have they checked the timing chain tensioners? The 320d are known to have issues with the chains, which could be causing your problems.
What maybe happening, is that as the car is left, the oil is running out of the hydraulic chain tensioner, causing a slack chain. When you come to start it, the oil pressure needs to build back up, before the tensioner tightens the chain enough, this can lead to the timing between cam and crank being wrong, hence the non start.
Certainly worth a look. Bad news though, the chain is down the back of the engine, so it needs to come out to be done, so also budget on a new clutch and DMF (especially if close to, or over 100k miles). If it has a full BMW history, and under 100k on the clock, you may be lucky and get some goodwill from BMW (although not very likely from what i am led to believe)
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