Both of my cars currently have the same issue. Brakes work fine AFTER you have pumped them a couple of times. They are not driven regularly (alas) and when I get in them the brakes flop uselessly to the floor when I press the pedal. Do this a couple of times and normal expected service is resumed. They have new cylinders all round and have been bled and adjusted. I suspect the master cylinder though there is no sign of leak (yes, I took the gearbox cover off to check properly, all dry, dry under the dust covers as well). Fluid level doesnt appear to be falling either.
In the absence of a better idea I have purchased a replacement master cylinder and will fit that, I have a couple of refurb kits so will refurbish the one removed... if there seems to be an improvement I will do the same for the second car, but it would be nice if someone said... 'yup, right thing to do, I had the same issue and replacing the master cylinder worked for me'. Its a bit of a faffy fiddly job otherwise (did someone once tell me the Morris Minor was a fantastic design? I love the cars but frankly the design has a number of very very poor decisions)
More brake 'issues'
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Re: More brake 'issues'
Sounds to me like a classic case of air in the system. New cylinders? Now matter how careful you were, it probably needs more bleeding. That said, I did have one car that had the same symptoms, and that was the MC. I'd still try more bleeding initially and change the MC as a last resort. You can't beat two person bleeding: open bleed screw, press and hold pedal, close bleed screw, release peddle. Repeat as needed. I've seen differing views as to whether the correct sequence is to start closest to the MC and work to furthest, or start at first and work to closest. The original workshop manual is somewhat reticent on the subject Don't just pick wheels at random.