Rather than give a step-by-step procedure, I thought I would share my knowledge of how the system works, and how I have adjusted it to work best!
The auto(matic) choke has a choke flap in the primary venturi of the carburettor (towards the back of the car). The position of this is set by a bimetallic spring coil, located under the cover. The operating linkage includes a cam with several steps. The thottle linkage has a long fast-idle-speed adjustment screw, which bears on this cam.
To 'set' the auto choke, you press the accelerator all the way to the floor, once. This causes the fast-idle adjustment screw to lift off the cam, allowing the bimetallic spring to position the cam (and choke plate) correctly. With the engine cold, the choke plate should be more or less fully closed at this point.
The plate itself is spring-loaded, so that with the engine running the choke will cancel itself to some extent. There is also a vacuum device that partially cancels the choke (a similar device is also fitted to manual-choke carburettors). The adjustment of the vacuum 'pull-down' is a procedure that requires measurement of the choke plate opening (using a twist drill bit) and is detailed in the Supplement chapter of the X1/9 Haynes manual.
As the engine warms, the coolant flowing through the choke housing will cause the bimetallic spring to cancel (open) the choke. However, the movement of the operating cam will be impeded by the fast-idle screw resting on each step. This is why the choke seems to be 'cancelled' by short stabs on the accelerator. It is also why the engine seems to idle fine for some time (the fast idle remains on the same step), yet once the car is driven, the engine stalls at the first intersection!
To check for coolant flow to the auto choke housing, I loosen the single 11mm nut in the centre of the housing cover, with the engine hot and running. There should be a leak of hot coolant.
You may wonder why the choke action (enrichment) is necessary. Without the extra fuel, the cold engine exhibits great hesitation as the throttle opens, and smooth progress becomes difficult. The secondary requirement of the choke is to raise the idle speed; the engine will not idle at the normal setting until quite warm. The 'fast idle' is often quoted at 1500RPM for setting with the engine warm, but remember that with a cold engine, the same throttle opening will give only 900RPM or less.
There are three settings for the auto choke: vacuum cancelling mechanism (see above), fast idle screw (adjustment either by throttle opening specification, or by trial-and-error), and the housing position (which affects both the initial amount of the choke, and the rate at which the choke comes off as the engine warms).
The Primary barrel idle jet also affects the choke operation. If this jet is too large (e.g. inscribed with '70' instead of the expected '47') then the choke will cause the carburettor to deliver too much fuel to the engine, giving a 'lumpy', pulsating quality to the engine idle.
If the auto choke 'cancels' too early, progress through the gears during normal driving will reveal flat spots at part-throttle. Of course, the engine may also refuse to idle as it begins to warm up (even slightly), yet may idle correctly once warm.
Do not be tempted to increase the fast-idle screw setting; since this may cause the engine to run too fast during initial starting. Also, the fast idle screw should not affect the idling once the engine is warm: if screwed in too far, it will.
If the idle speed tends to rise as the engine warms up, perhaps to 1500RPM, making reverse gear selection difficult, then the auto choke may not be cancelling early enough.
Mark the position of the housing before you make any adjustment, in case you have to return to the previous setting. I use correction tape.
Three slot-head screws secure the clamp ring of the housing. To gain access to the screws, you will probably need to slacken the 11mm centre nut and rotate the housing cover (with coolant pipes attached). Once all three screws are slack, temporarily re-tighten the cover nut, so that you can turn the housing slightly. With the air cleaner off, and the choke set, you should see the choke flap move.
To make the auto choke cancel sooner, rotate the housing clockwise. Do this if it seems that the engine revs tend to rise as the engine warms up., or that the engine idling tends to be lumpy (too rich).
Rotate the housing anticlockwise to give the auto choke a stronger effect (to overcome drivability problems), so that the choke flap closes fully.
After making any adjustment, remember to readjust the fast-idle screw the next time the engine is completely cold.
As the engine warms up, blip the throttle at regular intervals to check that the fast idle reduces in appropriate steps. Also check that the engine pulls as smoothly as can be expected...
Such adjustment of the choke will take a few days to perform and test under all conditions, since the engine must be cold each time.
Alternatively you may wish to remove the carburettor and perform the settings of the choke 'pull-down', the fast idle screw, and the choke housing position as per the measurements given in the supplement of the Haynes manual.