Hi NUTTER,
I had the exact same issue on my 03 M62. It was a blocked tranny cooler.
Easy to check. Drain your coolant and pop out the cooler from the plastic mount on the bottom of the radiator by sliding back the retaining clip and pulling it out. the oil lines can stay connected. Get your garden hose and stick it in each side of the cooler and see if you have a blockage. I'll bet you do.
On mine I had a full blockage so a coolant system flush wasn't going to help. I used a bit of coolant system acid cleaner in a squirty bottle to clean out mine until the new one arrived.
I also checked my Tranny thermostat and housing when I was checking for the issue with mine but it was fine and pretty clean inside.
When you get your new cooler I'll bet the oil line release tabs will brittle and you won't be able to get the oil lines off. A real PITA. To get mine off I used one of fuel line release plastic tool kits you get from supercheap but the tab that goes into the fitting is too short so you just grind down the outer part by about 5mm and it works a treat.
I also had the radiator out when I was doing my timing chains and although it was changed in 2016 there was quite a bit of brown flaky crap that came out of it when I flushed it. So I would recommend a good coolant system flush when you are putting it all back together.  
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