Suppose one paid a deposit and did not receive prescribed information confirming its protection within 30 days but only more than 8 months later. The landlord should be liable for 1-3x the deposit amount. How long does the tenant have to pursue this penalty from the landlord, and how could they go about suing for this?
Deposits are governed by Chapter 4 of the Housing Act 2004.
Within that chapter, the 30-day limit for sending the prescribed information to the tenant comes from section 213(6) (as amended):
(6) The information required by subsection (5) must be given to the tenant and any relevant person—
(a) in the prescribed form or in a form substantially to the same effect, and
(b) within the period of 30 days beginning with the date on which the deposit is received by the landlord.
An action to recover any sum recoverable by virtue of any enactment shall not be brought after the expiration of six years from the date on which the cause of action accrued.
Regarding making a claim if your deposit was not properly protected: this Shelter article describes in detail the steps involved.