If you leave before the fixed term ends there is no need to give any notice.
However, if you stay over, even by one day, a statutory periodic tenancy starts and you can only end that by giving notice to quit. That means, if you pay rent monthly, that you will be committed to at least another month; if you pay weekly then it will be four weeks. The really big snag is that you cannot give notice before the tenancy starts as section 5(5) of the Housing Act 1988 provides that a notice to quit given by a tenant before such a tenancy starts is of no effect. Giving notice on the first day of the periodic tenancy is (probably*) too late and so in effect you are committed to at least two months. If you want to stay for a short period, try and agree a further fixed term up to the date you want to leave, though note that the landlord does not have to agree to grant it. Until you have a clearer idea of when you will be leaving there is not a lot you can do.
*This depends on whether the day of service is included in the notice period and whether you you give notice ending the tenancy on the first or last day of a tenancy period.
so, they issue a S21, properly, but if you overstay you are now on a periodic and have to give effectively 2 months notice.
Surely the S21 makes the new periodic void?...how else could they apply to a court for eviction?
The law is an ass
Either they REALLY want the property back on the date the S21 calls, or they dont.