Just to add, the car seems to drive totally normally when I drive it, 4th is a little harsh when driving hard but my last E39 was the same so not sure if there's anything wrong there, My garage say the transmission oil level is ok as not long after I bought the car the coolant system was overhauled I'd been told a common problem so did it before it went wrong they did have to replace the gearbox oil cooling thing can't remember what it was called off the top of my head and they said they would have topped up the oil then, I'm fairly sure this was after the initial fault came up but I'll have to check my records.
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BMW E39 5-Series Transmission Fail Safe | 1997-2003 525i, 528i, 530i, 540i | Pelican Parts DIY Maintenance Article