At the time of writing (June 2022), the bill referred to in the White Paper has not been published. However, on p8 of the document (p12 in the PDF version), it states:
We know action is needed now and the Renters Reform Bill will bring forward
legislation in this Parliamentary session to deliver on our wide-reaching
commitments.
The current session began in May 2022, and is expected to run until around May 2023.
However, even if the bill completes its passage through Parliament within that period, changes of this nature usually don't come into force for some time, in order to allow landlords to change their plans.
I didn't spot a timetable for this in White Paper, but a useful comparison is a planned change to the Energy Performance Certificate (EPC) requirements for rental properties.
The government is proposing to raise the minimum EPC level for rental properties from grade E to grade C. It is currently just a proposal, but if it goes ahead, the current understanding of the timetable is this:
The Government’s proposal in the Bill is that rental properties meet a compulsory energy performance certificate rating of band “C” on new tenancies by December 2025, and on all rented properties by December 2028.
So 3 years from now for existing tenancies, and 5 for new ones. So by analogy, we might expect any change to "no-fault" evictions to follow a similar timetable, as a minimum.
We should know more when the bill is published.