3L diesels, which is 'best'?

No first hand but, apparently the Beemer variety are really cooking on gas these days. Sorry.

The 3 litre in the BMW is a great engine.
I think they're addressing the flammable EGR issue - mine's due to go in for its 2nd recall on this whenever I get round to booking it in.
Fuel economy on mine is sitting around 43/44 which is the same as the 320d that preceeded it.
It's a fairly smooth engine with a nice growl.

I don't have any experience of the Audi equivalent but sis spend a few months in a 67 plate A6 2.0TDi quattro.
The inside was a nice enough place but I got increasingly irritated by the bluetooth and it's lack of willingness to reconnect after even the shortest of stops, e.g. for fuel. The BMW isn't perfect in this regard but its certainly superior to the Audi.
I also didn't like the position of the heater and seat controls at the bottom of the centre console and angled slightly in at the bottom.
The other thing that I really didn't like was the gearbox. I don't know what's in the 3 litre but the 2 litre had the group's DSG. I found it very frustrating pulling out of junctions, there would be a slow pull away then a dramatic response after a slightly too long hesitation during which you'd pressed the throttle even harder.

Suffice to say I'm happier in a BM than I was in the Audi.

I have no experience of either, but do have plenty of experience of the Benz 3-litre which is wonderful. Smooth and quiet most of the time, with an almost V8 growl when you prod it. The 9G auto box is outstanding and never in the wrong gear. Reliability is good and my overall average since getting the car is 41 MPG. 47 MPG is normal for a motorway run. The E-class is very much a mile-eating cruiser, though, and not really a car for a B-road blast.

I very much like a 6 cyl diesel in the right car. I've driven the Porsche 3.0 d in the Macan and Panamera and it was very good and assume its just the VW lump in different tune and a better gearbox attached. I've driven a the Merc unit in an E-class with no complaints and been driven an X5 40d which certainly didn't want for performance.

SiL has the Audi A6 one. They’ve had it since new and it must be 6-7 years old. They do mega miles in it, between Czech, Austria and Slovakia on an almost weekly basis as well as holidays to France, Italy and Croatia. I think my nephew uses it for to and from Uni in Olomouc weekly as well, leaving the dodgy old Honda to his sister. It’s never done anything wrong and still looks smart.

The JLR 3.0 diesel in my car, don’t know the name, is refined but not entirely economical in that particular application. Then again in the XF Sportybrakes it was neck snapping quick and I don’t think that thirsty.

I ran the BMW twin turbo lump in the 640dGC - so same unit as found in all the *35d and *40d options.

It was / is a very impressive motor and, for a diesel, even made a reasonable noise/rumble. It was as fast as you could ever really need (perhaps not want although easy to map for more torque/power) and it was hugely flexible whilst delivering commendable economy on a longer run at steady speeds (around town etc nowhere near as good but that applies to almost anything).

If you are going the diesel route it's probably the best option out there and is very easy to live with and offers huge every day performance but (and this isn't specific to the BMW) it lacks character, interest and 'fun'. It was as fast as my old Jag SV8 and easily twice as economical but never raised a smile (more of a nod of respect)

My father had an A7 with the BiTurbo 3.0 diesel lump and I remember it sounded remarkably good for a diesel. He then had an SQ5 Plus, which I think had an even snortier version of the same engine. He liked them.