Yes. The Allotments Act 1950 covers everything and is designed to override any covenants or tenancy terms preventing you from keeping rabbits and hens (with caveats about welfare, H&S, and nuisance).
IANAL, but my understanding is that a byelaw may not overrule or contradict statute law, either.
12 Abolition of contractual restrictions on keeping hens and rabbits.
(1) Notwithstanding any provision to the contrary in any lease or tenancy or in any covenant, contract or undertaking relating to the use to be made of any land, it shall be lawful for the occupier of any land to keep, otherwise than by way of trade or business, hens or rabbits in any place on the land and to erect or place and maintain such buildings or structures on the land as reasonably necessary for that purpose:
Provided that nothing in this subsection shall authorise any hens or rabbits to be kept in such a place or in such a manner as to be prejudicial to health or a nuisance or affect the operation of any enactment.
12 Abolition of contractual restrictions on keeping hens and rabbits.
(1) Notwithstanding any provision to the contrary in any lease or tenancy or in any covenant, contract or undertaking relating to the use to be made of any land, it shall be lawful for the occupier of any land to keep, otherwise than by way of trade or business, hens or rabbits in any place on the land and to erect or place and maintain such buildings or structures on the land as reasonably necessary for that purpose:
Provided that nothing in this subsection shall authorise any hens or rabbits to be kept in such a place or in such a manner as to be prejudicial to health or a nuisance or affect the operation of any enactment.