We may need some more information about the whine to be able to take the diagnosis any further. Try telling us as much as you can about it. Is it high or low pitched? Just a whine, or any grinding sounds, or a noise like spinning ball bearings? Any rumble or knock? Does it happen when you bring the pedal up slowly, or quickly? Only on downward changes? Are you sure it is from the clutch area? You will understand that it is extremely difficult to guess the source of a noise from a long distance away. There will be lots of people along with various suggestions (most Mini owners have experience of clutch problems!), so first I thought we should eliminate what it is not, if that makes any sense.
Firstly, the hydraulics almost certainly have nothing to do with the whine. But don't mess about with slave cylinders of dubious origin. As they say, "buy cheap, buy twice". Just get the proper one from any Mini supplier, but only if it is leaking, or, less commonly, siezed. As you have not indicated that it has either of these problems, it should not need replacement. But don't ever swap new hydraulic parts into old cylinders. Much of the time the problem is bore wear and corrosion, and that will just chew up the new seal in a week or two. And the Allegro one may have a Metric bore diameter so the piston and seal will not fit the Mini body. (Not certain about that one way or the other).
You should always be referring to the Haynes manual (or the Rover manual), which gives the procedure for setting the stop nut, but basically you slacken off the locknut, wind the stop nut outwards a bit, get someone to hold the clutch pedal hard down to the floor, wind the stop nut up against the casing, then two more flats (120 degrees), and tighten the locknut. I am rather hoping that the problem is then cured, because if not, we have to start guessing, to a certain extent.
Your whine could be many other things, but I seriously doubt that it is the clutch. However, if you want to change something in the clutch, make it the thrust bearing, not the friction plate, pressure plate or flywheel. The fact is that the thrust bearing is the only part of the clutch that is likely to whine, and the only possibly whining part that is anywhere near easy to change. You already know how to take the clutch cover off, job almost done, as it is only fitted to the inside end of the central plunger. Beyond that, the problem would be drop gears (but they more commonly rattle, and whine at high speed) or gearbox.
By the way, the nuts are not 33mm, or any other metric size. I can't remember what they are, nowhere near 33mm, but it is an imperial size, like almost everything on a Mini, so you need a set of imperial A/F spanners, or you are sure to run into serious problems with damaged nuts or bolt heads sooner or later. In very few cases is the nearest metric size close enough to avoid damage. The Allegro was a metric car, which is why the banjo doesn't fit. The flywheel bolt is 1 1/2" A/F, which is 38.1mm.
I had a whine on changing down on a 998 some time back. It was there occasionally from new in 1984, and was due to the gearbox output shaft bearing, item 21 in the link below, having been sabotaged during manufacture. There was a neat angle grinder cut through the outer race, and that could not have happened by accident. It went quiet for a long while but eventually the sound returned and it locked up completely as I accelerated away from traffic lights. When it happened, only rarely, it sounded like a spinning dry ball bearing. Downright dangerous, because it allowed the final drive gears to wedge together (they were ruined), which is what locked the front wheels solid, but only under power. Quite rare for it to happen due to normal wear, I have never heard of it happening.
http://www.somerford...age=page&id=149
I hope that your problem is not that particular bearing failing because it is a lot of work to change it. However, I think that it is more likely at the other end. As I said, thrust bearing is the easiest, but if not, It may be worth having the transfer case off, and checking the drop gears and their bearings if nothing else is found. I generally remove the transfer case without lifting the engine, because it is quicker. No need to disconnect driveshafts, for example. But you do need to drain the oil. And, there is the sometimes difficult flywheel removal, which is needed whether the powerplant is still in the car or on the bench,
It is also remotely possible that the whine is due to failure of the primary gear bushes where it runs on the crankshaft. There is only relative rotation when the clutch is disengaged, it is all llocked up solid normally, and can't make a noise when the pedal is fully up. I would not describe that as a whine however, more like a grating noise, sometimes a squeal. Flywheel has to come off to check those properly. Red seal too, and the gear can be pulled out. But I don't think it will be those.
We are all here to help. You have given us quite a lot to go on, but any more that you can tell us will be useful.
.... for now anyway! If anything comes up again I'll definitely refer to this page as there is a lot of useful info 
