Hi Olaf
Always the first thing to do is to check the fluid level using Harvey's handy hints posted as a "sticky" at the top of the gearbox section.
Assuming all is OK there, then the most likely cause is that you have lost enough movement in the throttle linkage that the kickdown cable no longer has enough pull on it at full throttle. This is quite easily fixed and will also provide your engine with many extra horsepower! You don't say whether your car is a V8 or a 4 cylinder, so I shall assume V8. To start off have a look at these diagrams from the parts manual (even more useful than the workshop manual!).
http://www.rover-classics.co.uk/images/reference/thumbnailv8/throttle/index.htm
Looking at the diagram headed "Manual" the three components likely to be at fault are nos 25, 27 and 35. On the "Auto" diagram nos 34, 35, 44. See if you can identify these on your car and verify that they are present! The two bushes are frequently missing! The coupling often has a great deal of free motion.
It is also possible, but less likely, that the bushes supporting the acelerator cross shaft may be worn/damaged/missing. This would also affect the outcome.
With any of these components worn or damaged, the effect is that full throttle at the accelerator pedal fails to produce full throttle at the carburettors. The kickdown is operated by a cable that picks up on the accelerator shaft quite close to the carburettors. Therefore this cable fails to get a large enough pull to operate the kickdown.
All these bushes and couplings are available from Ian Wilson at Rover Classics and I presume from Winn's and JR Wadham.
After replacing any failed components you will need to set up the throttle linkage. Start by disconnecting the linkage from the carburretors and verify that the link between the carbs is set correctly. The carb furthest from the throttle linkage should reach full throttle at the same time as the carb closest to the linkage. Adjust the link between the two carbs until this is achieved. Ideally both carbs would now start to open at the same time also - you might have to compromise between the two criteria. Then reconnect the main linkage and verify that full throttle at the pedal equates to full throttle on the carbs. Adjust until this is achieved. You will have to wind off the idle speed adjustment and reset the idle speed and the idle mixture for both carbs. If you need to do this, post again and I will help you through this.
After this procedure it is almost certain that you will now have correctly operating kickdown. If you don't , or the change quality has suffered markedly, then I defer to Harvey as to how to adjust the kickdown cable. Note that the adjustment of the kickdown cable not only affects the kickdown, but also the preload of the change mechanism of the box. With it set wrong the box will be either too keen to change up or too keen to change down.
Hope that helps
Chris