as far as i am aware, once a correct Section 21 is issued, its efficacy is indefinite.
generally speaking the local authority will not rehouse someone if they are not in immediate danger of being evicted - the only proof of that is a possession order from the courts -
i think you will have a long wait if you expect the tenant to pay for their own eviction - there is no incentive whatsoever for them to do so ......
Clottie The Positive “Windswept and interesting”
The Somerset-Lancashire lady
Aviatrix extraordinaire !