Simple answer is that the tenancy of your plot and the membership of your site association and two separate things. Only they're not entirely separate.
On the one had there's the allotment tenancy, and the tenancy agreement will contain terms that say what you can and can't do, like you can't sunbathe naked, and you must paint your shed turquoise, and crucially it can contain a term that says you must be a member of the allotment association. You're also right that as a consequence of the Allotments Act 1922 as ammended by the Allotments Act 1950 you can't be given less than 12 months notice to quit. As it happens that's not important, what makes the difference is Section 146 Law of Property and the term of the agreement that deals with forfeiture.
Then on the other hand there's the membership of the allotment society, which is actually a contract for a service. Your membership will also be subject to terms such like you can't be a member if you don't have an allotment, or that you can be expelled from membership with five days notice if you bring the association into disrepute.
There's not actually any incompatibility. Say you insult the mayor infront of the television cameras at the opening of your new site toilet, that's probably enough to get you expelled from the association for bringing it into disrepute. Five days later you're not an association member any more, and so you are now in breach of a term of your tenancy agreement - but you are still a tenant. What the committee has to do now is forfeit your tenancy for breaching the term about being an association member - that's not a notice to quit. The statutory procedure for forfeiture is that if you can't re-join the association then the committee can forfeit the tenancy immediately.
Forfeiture and Notice to Quit often get confused but they have absolutely nothing whatsoever to do with each other, except of course they both end the tenancy.
Forfeiture, which is governed by Section 146 Law of Property 1925, is where the tenancy is brought to an end because of some breach of a condition of tenancy.
Notice to Quit is a bit more subtle. Technically a lease is called a a term of years absolute. What that measn is that it actually runs for a definite fixed period. Any fixed period actually, not just a year, it could be a day or a thousand years, but it must be a definite fixed period. But it happens that when people want to lease land they don't always know how long they'll want it for, and so with a bit of legal smoke an mirrors the periodic tenancy was invented, and allotments are almost always let on an annual periodic tenancy. With a periodic tenany the lease is still for a definite fixed period, like a year for an allotment, but rather than terminating at the end of the year another year's tenancy is automatically appended - glued to the previous one with some special legal glue so that looking back you can't see the join! A Notice to Quit is the magic incantation that stops the periodic tenancy re-starting with a new period - and this is why when a Notice to Quit is served it must expire on either the last day of the period or what would be the first day of the next period, because it isn't actually bringing the tenancy to an end, it's just preventing it from starting up again automatically.
A bit complicated I know, but I've seen tenancy agreements that talk about serving a 30 day Notice to Quit for breaking the rules, and it's complete rubbish because what they actually have to say is a Notice of Forfeiture, and because the rule purports to allow the tenancy to be ended with a 30 day Notice to Quit the allotments acts say that's void, and because there is no implied common law right to forfeit a tenancy the landlord is stuffed.
Oh, as always, this is all IMHO and it could be completely wrong. Just saying.
