The code I got back was 271D, as in pic above, which is the up-stream or 'front-of-cat' sensor heating. Codes 271A and 271B show fault at the same sensor, but are labelled directly 'front-ofcat-sensor' with no heating suffix. This sensor is located slightly forward of the catalytic-convert mounted about halfway down the car's underside, as in the diagram below. The one shown to the rear of the cat is the down-stream or 'rear-of-cat' sensor. Code 271C applies to this rear sensor, along with 271E for Rear O2 sensor-heating. The two sensors, though essentially the same, are different parts and part numbers due to the rear sensor having a longer wire. [List of emission based fault-codes here.]
I got hold of a used sensor taken from another E46 318i from www.marobmwspares.co.uk for £18. I prefer used OWM BMW parts that come with a warranty / guarantee over cheap spurious items. The least expensive spurious items start at about £20 for a sensor that can be used for either front or rear of cat, having a long wire than can be cut to length with two bare wire or universal wire-terminal ends that require the original plug to be soldered on from the broken sensor. On the other end of the spurious price range was Euro Car Parts, who offer an OEM-quality sensor for £68, which is pricey but comes with the correct wire-length and plug fitted.
My old sensor proved very tricky to remove with the car on fixed-ramps and I would not like to attempt it with jack and axle-stands. I was unable to crack the sensor off and unscrew it even with the use of special Lambda-sensor sockets and it began to round so I ended up booking ramp-time in the workshop of the company where I work and lifted the E46 Touring on a 2-post ramp to give loads of access underneath. To avoid the risk of breaking the Lambda-sockets or further rounding the sensor itself, I decided to cut about half of the sensor off using a 3.5" cutting-disc and used a 22mm deep impact-socket with a long breaker-bar to finally crack the sensor off. It was pretty seized on, but once cracked just unscrewed by hand. The replacement sensor screwed straight in and swapping / mounting the plug was easy to make for a 5-10-minute job once the car was positioned on the ramp legs.
RESULTS:
The emission test result went from 2.2, with a maximum of 2.0 allowed, down to just 1.0 with the faulty sensor replaced. There is also a significant drop in neat-petrol smell emitting from the rear of the car when it is started from cold, so I'm glad this problem showed itself and got fixed.
Article on /BMWTech subreddit: https://www.reddit.com/r/BmwTech/comments/d5v30t/271d_lambda_sensor_front_of_cat_which_one_there/