Does that alteration to the Allotments act apply to home gardens as well?
Yes. The Act doesn't place any restrictions; it says (my emphasis):
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, ...
That being said, many people who are in a position to enforce rules relating to use of land (e.g. landlords) are not aware of this provision in the Act and may make life difficult for the chicken keeper or take indirect enforcement action through other routes.
So if, for example, I was renting a house to live in, I wouldn't get any chickens in the garden unless I had the permission of the landlord, because whilst this Act allows you to have them, the landlord can always give you notice to leave on a no-fault basis - i.e. for any reason - once the Covid19 restrictions on doing so have been removed / relaxed.
On the other hand, if I'd bought a house and there was something in the deeds which said no chickens, in law that clause is trumped by this Act because you can't sign away your statutory rights.